How To Find Limits By Looking At A Graph

Finding Limits Graphically

By the stop of this lecture, you should be able to use the graph of a function to find limits for a number of different functions, including limits at infinity, and to decide when the limits practice not exist (and when they do non be, to explain why). Y'all should also be able to utilize limit annotation correctly.

Considering this lecture focuses on limits, allow'due south review our breezy definition of limits from the last lecture:

Limit (breezy definition)

If f(x) somewhen gets closer and closer to a specific value L as ten approaches a chosen value c from the correct, then we say that the limit of f(ten) as x approaches c from the right is 50.

If f(10) eventually gets closer and closer to a specific value L as x approaches a chosen value c from the left, then we say that the limit of f(x) every bit x approaches c from the left is L.

If the limit of f(x) as ten approaches c is the same from both the correct and the left, then we say that the limit of f(x) as ten approaches c is L.

(If f(10) never approaches a specific finite value every bit x approaches c, so we say that the limit does not be . If f(x) has unlike right and left limits, then the two-sided limit (lim_{x →c} f(x)) does not exist.)

Notation:
Specifically, we write:

• lim_{x →c-} f(x) = L to denote "the limit of f(x) every bit ten approaches c from the left is L"

• lim_{10 →c+} f(x) = Fifty to announce "the limit of f(x) as x approaches c from the correct is L"

• lim_{10 →c} f(x) = L to denote "the limit of f(x) as x approaches c is L"

Exist conscientious!: The plus or minus sign which appears later on the c denotes the direction from which ten approaches c - it does NOT mean that c itself is positive or negative (information technology may be either)!
For example:

• lim_{10 →-2-} f(10) = L means that the limit of f(ten) every bit ten approaches -2 from the left is L;

• lim_{10 →-2+} f(x) = Fifty means that the limit of f(ten) as ten approaches -2 from the right is Fifty;

• lim_{x →2-} f(x) = L ways that the limit of f(10) as x approaches 2 from the left is L; and

• lim_{10 →2+} f(10) = Fifty means that the limit of f(x) as x approaches ii from the right is Fifty;

This definition is informal because we haven't formally defined what nosotros mean past "approaches" or "somewhen gets closer and closer". Once nosotros accept developed a more intuitive feel for what limits really are, we volition come back and formally define these terms.

The all-time way for us to better understand what a limit is, is just to jump in and start looking at unlike functions and discussing what the limit might exist for specific x-values for that function. But earlier we exercise that, nosotros briefly innovate a definition that nosotros will use in the examples that follow. Throughout this lecture, nosotros will often use the word discontinuity , so we define it (at to the lowest degree informally, for now) hither:

Continuous, Discontinuity (informal definition)

A function is continuous in an interval if the graph of that part over that interval tin be drawn in ane stroke (without lifting your pen from the newspaper).

A discontinuity on a graph is whatever indicate at which the graph is NOT continuous (e.k. a hole, a jump, an asymptote).

In that location are more formal definitions of continuity, but we use this one for at present because it is easy to sympathise intuitively. Later nosotros will come up back and ascertain this concept more formally.

Limits of Specific Functions

The all-time way for the states to empathise what limits really are is to expect at a bunch of different examples which exhibit different types of behaviors around x=c for some fixed value c. Then, let's jump correct in!

In this lecture, we work out each instance only past looking at the graph. However, each of these functions can besides be expressed algebraically (with an equation) and we tin likewise find the limits of functions algebraically by using this equation to calculate the limit. Nosotros volition talk virtually how to do that in the next lecture.

A simple case, where lim_{x →c} f(x) = f(c):

For many straightforward functions, the limit of f(ten) at c is the same as the value of f(x) at c . For instance, for the function in the graph below, the limit of f(x) at 1 is simply 2, which is what we get if we evaluate the function f at 2. Considering the bespeak (1,2) is on the graph of f(10), the limit is is 2, so nosotros could write:

lim_{x →i} f(x) = 2

This example wasn't very interesting, considering information technology doesn't actually make it clear why we even need to calculate a limit here here: considering this function is completely continuous effectually x=1, the value of the office at i and the value that the function approaches as x gets closer to ane are the same thing!

The limit definition, however, is specially useful for functions where the function is non defined exactly at x=c (but where it is divers all around c ), or where the value of f(x) at c is different from the value that f(x) approaches every bit ten approaches c.

An example with a hole at x=c:

For the function in the graph beneath, f(10) is non defined when x = -2. Looking at the graph, nosotros can see that at that place is a hole, or discontinuity, at that point. However, in this case, nosotros can meet that equally we motion forth the line representing the function f(x) from the left towards x = -two, the value of f(x) gets closer and closer to -4. Similarly, as we move forth the line representing the function f(x) from the right towards x = -ii, the value of f(10) gets closer and closer to -4. And so in this example, we can conclude that:

lim_{x → -2} f(x) = -four

An instance with a function that has a leap discontinuity at x=c consisting of a single point:

For the function in the graph below, f(x) is defined when x = -ii, merely the value of f(x) at -2 is non at all similar to the value which f(x) will approach as x gets closer to -two from either the left or the right. Looking at the graph, nosotros can see that in that location is a spring discontinuity at that bespeak so that when x = -two, f(x) = -i; nevertheless, when x is almost (but NOT equal to) -2, f(x) is actually close to -4.

In this instance, nosotros tin can see that as we motion along the line representing the function f(x) from the left towards ten = -two, the value of f(x) gets closer and closer to -4. Similarly, as nosotros move along the line representing the function f(x) from the right towards x = -ii, the value of f(10) gets closer and closer to -4. So even though the function actually equals -1 when nosotros are actually at ten = -2, in every other point effectually 10 = -ii, the function is approaching -4 instead. So in this case, the limit is really different from the function value at that betoken:

f( -two) = -ane ; but

lim_{10 → -two} f(x) = -iv

An example with a office that has a bound discontinuity at x=c, and different limits from the right and from the left:

For the office in the graph below, f(x) is defined when x = 1, but the value which f(10) will arroyo as x gets closer to 1 from the left is different from the value that it will approach as x gets closer to 1 from the correct. Looking at the graph, we can see that as x approaches i from the left, f(x) approaches negative two; yet, every bit x approaches 1 from the right, f(x) approaches positive 2.

Notice again that the bodily value of the function at 1 is non relevant to finding the limit: it is possible that sometimes the limit of f(ten) at c will actually be f(c) (as happened in our commencement case in this lecture), but much of the time, the limit of f(x) at c will be dissimilar from the value of f(c), specially when f(c) is not defined, or when there is a aperture at x=c.

In this instance, lim_{x →ane+} f(ten) = f(c), but lim_{ten →1+} f(x) does not equal f(c). Also, because the limits are different from the left and from the right, the two-sided limit lim_{ten →1} f(x) does not exist. Specifically, nosotros can write:

lim_{x →i-} f(x) = -2

lim_{x →ane+} f(x) = 2

lim_{10 →1} f(x) does not exist

An instance with a function that has an infinite aperture (or vertical asymptote) at x=c:

For the function in the graph below, f(x) is non defined when x = 0 because as x gets closer and closer to 0 from either side, f(x) but keeps getting bigger and bigger: the closer that x gets to 0, the bigger f(x) gets. At ten=0, f(x) doesn't accept whatever specific value on the graph.

When this happens, we say that f(x) increases without jump as 10 approaches c. Every time this happens with a limit, nosotros can only write that the limit does not exist because f(x) does not arroyo whatever fixed, finite value as x approaches c. However, that is a niggling unwieldy to write, so instead mathematicians accept invented a fleck of shorthand for this:

Note for limits that increase (or decrease) without bound, ±∞

Notation:

• If f(x) increases without bound as x approaches c from the left, we can write:
  lim_{10 →c-} f(x) = ∞ or lim_{ten →c-} f(10) = +∞

• Similarly, if f(x) increases without bound every bit x approaches c from the right, we tin write:
  lim_{10 →c+} f(x) = ∞ or lim_{x →c+} f(10) = +∞

• If f(10) increases without spring equally x approaches c from both the left and the right, we can write:
  lim_{x →c} f(10) = ∞ or lim_{x →c} f(x) = +∞

At that place is also the possibility that f(10) will decrease without spring (get smaller and smaller, or more than and more negative) as x approaches c. In that case, nosotros can utilise the symbol -∞.

Be conscientious!: The use of the infinity symbol ∞ is just a shorthand for maxim that the limit does Not exist because it increases or decreases without bound.

But infinity, or the symbol ∞, is Not a number, and then no equation or part can ever actually be equal to infinity!

We must be very careful how we use the infinity symbol: we tin can say that a limit is equal to infinity to indicate that the beliefs of the office as ten approaches c "blows up" indefinitely, but nosotros can NEVER say that f(x) itself is equal to infinity, because no function can ever attain a value of infinity: at every single bespeak on the graph of f(ten), f(x) has a specific value, even if that value is very very big (or very very pocket-size). (Even if f is undefined at a specific point, it is not equal to infinity at that bespeak - if it is undefined, and so it is not equal to anything at that signal.)

So, returning now to our example, nosotros can write:

lim_{x →0-} f(10) = + ∞

(or, the left-sided limit does non exist because f(ten) increases without bound)

lim_{x →0+} f(x) = + ∞

(or, the correct-sided limit does non exist because f(x) increases without bound)

lim_{x →0} f(ten)= + ∞

(or, the tow-sided limit does not exist because f(x) increases without spring)

An example with a function that has an space discontinuity (or vertical asymptote) at 10=c, with different limit behavior from the left and from the correct:

For the part in the graph beneath, f(x) is non defined when x = 1 because as ten gets closer and closer to one from the right, f(x) but keeps getting bigger and bigger: the closer that x gets to one from the right, the bigger f(10) gets. And as x gets closer and closer to one from the left, f(x) just keeps getting smaller and smaller (or more and more negative): the closer that x gets to 1 from the left, the smaller (or more negative) f(x) gets.

And so at x=i, f(ten) doesn't have any specific value on the graph.

And then in this case nosotros tin write:

lim_{x →1-} f(10) = - ∞

(or, the left-sided limit does not exist because f(10) decreases without bound)

lim_{x →one+} f(x) = + ∞

(or, the right-sided limit does non exist because f(x) increases without jump)

lim_{ten →1} f(x)= ∞

(or, the ii-sided limit does non be because f(x) decreases/increases without jump)

We've now seen two examples where f(x) has increased (or decreased) without jump as x approached a specific value c, merely the reverse type of behavior can besides occur: We can consider what the behavior is of f(ten) when x increases (or decreases) without leap. In some cases, information technology may happen that as x gets bigger and bigger, f(10) gets closer and closer to a specific value Fifty, and in those cases, we could write something like this:

lim_{ten →+ ∞} f(x)= L

Similarly, we could consider what the behavior is of f(ten) when ten decreases without bound, and if f(ten) were to arroyo a specific value L, nosotros could write something like this:

lim_{x → - ∞} f(x)= L

Now we motion on to some examples where we consider the behavior of f(x) as x increases or decreases without bound. In other words, we try to determine if f(x) approaches a specific fixed finite value as we go further and further to the right, or the left, on the graph of f(ten).

An example with a function that has a limit of zero at infinity:

For the office in the graph beneath, we offset consider the beliefs of f(x) equally every bit x increases without bound, or in other words, we consider what happens to f(x) every bit we move farther and farther to the right on the graph. In this case, f(x) appears to become closer and closer to zero. It tin can never reach nada, considering the function has no end: x tin can continue to increment forever. Only the behavior of the function equally x increases is that it grows always closer to 0, even if it can never reach it.

Similarly, as x decreases without leap, or as we move farther and farther to the left on the graph, f(x) appears to become closer and closer to zero. Again, hither the behavior of f(10) as x decreases (or grows more than and more than negative) is that it grows ever closer to 0, even if it can never accomplish information technology.

So in this example we can write:

lim_{10 → +∞} f(ten) = 0

lim_{x → - ∞} f(x) = 0

An instance with a part that has a limit of two at infinity:

For the role in the graph below, we beginning consider the behavior of f(ten) as as x increases without bound, or in other words, we consider what happens to f(ten) as we movement further and farther to the right on the graph. In this case, f(x) appears to go closer and closer to two. It can never reach two, because the part has no end: x can go along to increase forever. But the behavior of the function every bit 10 increases is that it grows e'er closer to ii, fifty-fifty if it tin can never reach it.

Similarly, as x decreases without bound, or as we motion farther and farther to the left on the graph, f(x) appears to get closer and closer to two. Again, hither the beliefs of f(x) as x decreases (or grows more and more negative) is that it grows ever closer to 2, even if it can never achieve information technology.

So in this case nosotros can write:

lim_{x → +∞} f(10) = 2

lim_{ten → - ∞} f(ten) = 2

An example with a function whose limit does not exist at infinity:

For the office in the graph below, we get-go consider the behavior of f(x) equally every bit x increases without bound, or in other words, we consider what happens to f(x) as we motility further and farther to the correct on the graph. In this example, f(x) appears to increase without bound: it just seems to get bigger and bigger equally we motility to the right on the graph, without always approaching a specific y-value.

Similarly, equally ten decreases without bound, or as we move farther and farther to the left on the graph, f(x) appears to subtract without leap: information technology just seems to get smaller and smaller (or more and more negative) as we movement to the left on the graph.

So in this instance we can write:

lim_{x → +∞} f(x) = + ∞

(or, the limit does not exist because f(x) increases without spring)

lim_{x → - ∞} f(x) = - ∞

(or, the limit does not be because f(x) decreases without spring)

We note that it need not be the case that f(x) increases without bound equally x increases without jump and that f(x) decreases without spring as ten decreases without bound. For example, for the function in the graph below, we would have the opposite:

lim_{x → +∞} f(x) = - ∞

(or, the limit does not exist considering f(x) decreases without bound)

lim_{x → - ∞} f(x) = + ∞

(or, the limit does not exist because f(10) increases without bound)

An case with a function that has unlike limits at positive and negative infinity:

It is besides possible for a function to have limits at positive infinity and at negative infinity that are unlike (or even for the limit to exist for one of these, but not for the other). All this statement really ways is that the beliefs of f(x) could be very different on the far left of our graph than on the far right. For instance, in the graph beneath, we can see that a limit does be both every bit x decreases without bound and equally x increases without bound, and that this limit is different in each case:

lim_{x → +∞} f(ten) = 1

lim_{x → - ∞} f(10) = -1

There is no reason why our limit would demand to be negative equally x becomes "more negative" (i.e. as 10 → - ∞) or that it would demand to be positive as x becomes "more positive" (i.eastward. every bit x → + ∞). For example, we could accept the opposite case, as we do in the function given in the post-obit graph:

lim_{x → +∞} f(x) = -1

lim_{x → - ∞} f(x) = i

An instance with a function that has an aquiver discontinuity:

I concluding possibility when we await for a limit as 10 approaches c is that f(ten) never approaches annihilation every bit x gets closer and closer to c: we could take beliefs, similar in the graph given below, where f(x) just oscillates more and more wildly as x gets closer and closer to c from either the left or the correct.

With the motion-picture show of the graph higher up, it's difficult to be sure exactly what is happening as x gets closer and closer to 1: it looks similar the oscillations are getting more and more than dumbo, so that f(ten) is continuously moving back and along between -i and 1 without ever settling down, but information technology is impossible to be certain. (Without an bodily equation to look at that would explain exactly what the values of f(x) are as we get closer to ane, we really can't testify that this is the behavior of this particular function, and so for now, I'll just enquire you to take my word for it.) But to go a ameliorate idea of how this beliefs of f(x) really works as x approaches 1, you can experiment a bit with the interactive animation below.

In the interactive animation below, you can see this behavior more conspicuously by moving the slider to the correct, which zooms in the ten-values on the graph around one. We can run into that as we zoom in effectually one, the graph just oscillates more and more oft, until it is so dense that nosotros can no longer come across the spaces between the graph and the bare infinite around information technology.

zoom

In this item example, the only determination that we can brand (assuming that you lot are convinced that this part has the behavior that I have described and so far in words), is that the limit of f(10) at ten=c does not be. In this case, the only thing that we can write is simply that:

lim_{x → one} f(x) does non be

In this instance, nosotros can NOT write that the limit is equal to infinity, considering the behavior of f(ten) as 10 approaches 1 is NOT that it increases without bound - rather, the beliefs of f(10) equally x approaches 1 is to oscillate indefinitely among -1, one, and all the numbers that fall in between.

This is important, considering it is essential for u.s. to understand that limits may neglect to exist for different reasons. We can only write lim_{x →c} f(x) = ± ∞ in cases where the limit fails to be because f(x) increases or decreases without leap, and NOT in cases where f(x) has different values on the left and on the right, NOR in cases where f(x) oscillates indefinitely among a set of values as 10 approaches c.

So far, we have discussed all the ways in which nosotros tin can find limits graphically, by considering the graph of a role visually. In the next lecture, we volition give equations for each of the functions presented in this lecture, and we will explore how we can notice limits of functions algebraically (even if we can't look at the graph).

But before we end this lecture, let's take a moment to remember about the basic patterns and structure nosotros observed by looking at this collection of dissimilar functions with very different types of beliefs effectually ten=c.

One of the most important skills in mathematics is the ability to recognize patterns and to be able to categorize expressions, equations, or other mathematical objects based on their properties.

Mathematics is about looking for construction (categorizing equations, expressions or other objects based on common backdrop)!

For example, when we practise homework issues, nosotros may work out thirty different bug, just unremarkably many of the problems have the same construction, even if they have different numbers or variables. Oftentimes those 30 bug could really be categorized into something like five dissimilar types of problems. If we can see that, so we can make our lives a whole lot easier, because now instead of trying to do 30 problems, we only need to sympathize the arroyo needed to solve problems of each of the five types.

Hither is an example of a problem that many algebra students struggle with, usually considering they get distracted by details and variables and forget to look for the underlying structure of the equation:

Solve for h: A = 2Πrh + 2Πr ²

When aiming to solve this problem, many students go distracted past the fact that the equation contains a lot of variables, and a square, and the irrational number pi. However, none of these details are actually relevant to the structure of the equation. This equation really just has the following construction:

A = Bh + C

where B and C just stand in for "some expression that doesn't contain any h's". In other words, we don't really care how many odd or interesting algebraic expressions are written in the place where 2Πr appears in the original equation - instead we can only recall of this whole expression equally merely the coefficient of h. All we care almost here is that in order to solve for h, we need to be able to get h by itself on one side, and and then we need to subtract any expression on the right that does not contain h from both sides, and so that we tin can eliminate it on the right side of the equation. And then once we've done that, we will need to divide both sides of the equation by anything that is a coefficient of h.

But all of the following examples have the same structure as the original trouble to a higher place:

Solve for h: A = 2Πrh + 2Πr ²

Solve for h: A = Bh + C

Solve for h: 42.6578 = 94h + 0.55

Solve for h: (Πr ^two A)² = Πr ⁶ A ⁷ + (2Πr+ 2Πr ^two)h

Solve for h: ane = 2h + 3

Solve for h: 1 = (three+4r) + 2h

Each of these examples says that some quantity is equal to some other quantity times h, plus another quantity. And then to solve each of these equations, we will simply need to subtract the term which does non contain h from the both sides of the equation (to cancel it out to cipher on the right-manus side), and then we will need to divide both sides by the full coefficient of h (i.e. whatever is beingness multiplied by h, all the same messy and complicated it may be).

So, how does this help u.s.a. to ameliorate sympathise what is going on in calculus (or other math classes that are not algebra)? In this case, it makes sense to think back about all the limit examples we have merely looked at in this lecture. Why did I choose the exact examples that were covered in this lecture? I didn't choice them at random. I specifically selected them and then equally to cover every possible kind of behavior that we might discover effectually x=c, and then as to give you a expert experience for all of the things that can happen when we are trying to evaluate a limit. So permit's go dorsum over the examples that nosotros merely covered in this lecture, and meet if nosotros can look for underlying construction and patterns.

How many unlike kinds of behavior can a office f(x) have effectually x=c?

1. f(x) is continuous in an interval around x=c. In this case, the ane-sided limits from the right and the left are the aforementioned, and they are both equal to f(c). (For case, in the graph below when c=1.)
   
2. If f(10) is non continuous in whatsoever interval effectually x=c, then one of the following things could happen:
   1. There is a pigsty at 10=c only at that place is an interval around x=c so that f(x) is continuous everywhere in that interval except at x=c. In this case, f(x) is undefined at 10=c, but the one-sided limits from the correct and the left are the same, and they are both equal to f(c). (For instance, in the graph below when c=-2.)
      
   2. There is a spring at the single point where x=c , merely there is an interval around x=c so that f(x) is continuous everywhere in that interval except at 10=c; then the one-sided limits from the right and the left are the aforementioned in this instance, but they are not equal to f(c). (For example, in the graph below when c=-2.)
      
   3. There is a jump at the signal where x=c , only there is an interval around x=c so that f(x) is continuous everywhere in that interval except at x=c; even so, in this instance the i-sided limits from the right and the left are different, and at least one of them is not equal to f(c). (For example, in the graph below when c=ane.)
      
   4. There is a vertical asymptote at x=c , so that f(x) increases or decreases without bound as x approaches c from the right and/or the left. In this instance, neither the ane-sided right-mitt or left-hand limits exist, and so the two-sided limit does not exist either. (For example, in the graph below when c=1.)
      
   5. In that location is some sort of "actually unusual" behavior around x=c , such as increasingly more than dumbo infinite oscillations and then that f(x) never approaches any specific value every bit x approaches c (either on one side only, or from both sides). (For example, in the graph below when c=ane.)
      
      Another thing that could happen (which is not covered past an case in the previous lecture) is that f(x) could be discontinuous everywhere in some interval around 10=c. This is difficult to draw graphically, but we can think of an example where we ascertain a function in a manner so that information technology jumps around at every single point on the real number line. For example, a function called the Dirichlet function is discontinuous at every single signal. Here is how information technology is defined: If x is a rational number, then f(x)=1; but if x is an irrational number, f(ten)=0. Agreement why this function is discontinuous everywhere is a bit beyond the telescopic of this course, but if you lot are interesting in understanding why that is and then, you lot simply need to take the premise that around whatsoever given irrational number, nosotros cannot find an interval small enough to exclude all rational numbers; and around any given rational number, we cannot discover an interval minor enough to exclude all irrational numbers. This means that around whatsoever given x=c value, we can never find an interval around c where f(x) is continuous (even if nosotros allow f(x) to be discontinuous at x=c).

For limits at infinity (i.due east. equally x approaches ±∞), the situation is less complicated. There are essentially two possible cases:

1. As 10 approaches ±∞, f(ten) approaches a specific finite value 50, in which case the limit is Fifty. (This is the same thing as proverb that f(10) has a horizontal asymptote at ten=c.)
2. As x approaches ±∞, f(10) never approaches a specific finite value, but rather just keeps increasing or decreasing indefinitely in an unbounded manner (i.e. f(x) increases or decreases without bound, or approaches ±∞). In this instance the limit does not exist.

Equally we move forrard into the next lecture, where we will aim to find rules and procedures for calculating limits algebraically (i.e. using just the equation for f(x) instead of the graph), we will want to keep this listing of the possible types of limit behavior in listen! This will help us to figure out how many unlike types of algebraic techniques nosotros may need to use in order to evaluate limits - if two different functions have different types of behavior around x=c, and then nosotros may have to use 2 different methods to finding the limit for each role algebraically!

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