14.01 Limits from graphs and tables of values

Introduction to limits

Many functions display trends as their domain gets very large or very small - they may get closer to a single value, or become larger and larger.
Additionally, some functions are undefined at particular input values - for example, the function $\frac{1}{x^2}$1x2​ when $x=0$x=0, or $\tan x$tanx whenever $\cos x=0$cosx=0 - and the behavior of such functions for inputs near these values is often quite interesting.

In order to talk about these trends, when inputs get close to a breaking value, we use a concept called a limit. Luckily for us, we only need this single concept to discuss each of these scenarios in a consistent way.

 

Long-term trends in functions and horizontal asymptotes

A function can behave in a few different ways as $x$x becomes very large or very small - we need to consider both the positive and the negative directions of "large". Some functions, like $f\left(x\right)=x^2$f(x)=x2, become larger and larger the further away the input is from the $y$y-axis. We can tell this from the graph of the function:

This is denoted as follows:

$\lim_{x\rightarrow\infty}x^2=\infty$limx→∞x2=∞ (say the limit as $x$x approaches infinity of $x$x squared is infinity)

and

$\lim_{x\rightarrow-\infty}x^2=\infty$limx→−∞x2=∞ (say the limit as $x$x approaches negative infinity of $x$x squared is infinity) 

to express that the function values become arbitrarily large and positive as the input values trend towards infinity.

Others, like $\frac{1}{x^2}$1x2​, become smaller and smaller, trending towards a finite value - in this case $0$0. We can see this from its graph as well:

In both the negative and positive directions, the graph becomes closer to the $x$x-axis as $x$x increases.

We write

$\lim_{x\rightarrow\infty}\frac{1}{x^2}=0$limx→∞1x2​=0 and $\lim_{x\rightarrow-\infty}\frac{1}{x^2}=0$limx→−∞1x2​=0

to express that the function values become arbitrarily close to $0$0 as the input values trend towards infinity. Importantly, this function looks more and more like the line $y=0$y=0 the further from the axis we travel. This line is a horizontal asymptote for the function, and is usually drawn as a dashed line.

Another example of a function with a horizontal asymptote is $f(x)=\frac{x^2}{x^2-1}$f(x)=x2x2−1​.
If we put a very large number (either positive or negative) into this function, the answer will be very close to $1$1. In fact, the bigger the input we choose, the closer the output will be to $1$1. We therefore say that $y=1$y=1 is a horizontal asymptote, and write

$\lim_{x\rightarrow\infty}\frac{x^2}{x^2-1}=1$limx→∞x2x2−1​=1 and $\lim_{x\rightarrow-\infty}\frac{x^2}{x^2-1}=1$limx→−∞x2x2−1​=1.

We draw in the horizontal asymptote on the graph as follows:

These two types of trends (to infinity, and to a finite value) can occur in the same function, depending on whether we are trending towards $\infty$∞ or $-\infty$−∞, such as

$\lim_{x\rightarrow\infty}e^x=\infty$limx→∞ex=∞ and $\lim_{x\rightarrow-\infty}e^x=0$limx→−∞ex=0.

We still say that $f(x)=e^x$f(x)=ex has the horiztonal asymptote $y=0$y=0 even though it only trends towards this line in one direction.

But what about a function like $f\left(x\right)=\sin x$f(x)=sinx? We know from the graph of $\sin x$sinx that its output is always between $1$1 and $-1$−1, so the limit is not infinite in either direction. But it also never "settles down", trending closer and closer to a single value, in the same way as the other examples with finite limits we have seen already:

The function does not look more and more like any horizontal line as $x$x becomes large - it does not have an asymptote, and we say that the limit $\lim_{x\rightarrow\pm\infty}\sin x$limx→±∞sinx does not exist. 

 

Undefined values and vertical asymptotes

We have seen limits of functions as the input value $x$x becomes very large. But the notion of a limit works just as well for finite values of $x$x. Most finite limits for most functions don't tell us any more than we already knew - for example, the limit

$\lim_{x\rightarrow2}$limx→2  $x^2$x2

can be read as "the value of $x^2$x2 as  $x$x approaches $2$2". This is $2^2=4$22=4 - the value we get when we substitute $2$2 into $x^2$x2. The real power of limits is their ability to assign values to expressions like

$\lim_{x\rightarrow0}\frac{\sin x}{x}$limx→0sinxx​.

If we simply substitute $x=0$x=0 into $\frac{\sin x}{x}$sinxx​ we get $\frac{0}{0}$00​, showing that the function $f\left(x\right)=\frac{\sin x}{x}$f(x)=sinxx​ is undefined for $x=0$x=0. However, examining a table of values for the function reveals that approaching $0$0 brings us closer and closer to a particular function value:

$x$x	$-0.1$−0.1	$-0.01$−0.01	$-0.001$−0.001	$0$0	$0.001$0.001	$0.01$0.01	$0.1$0.1
$\frac{\sin x}{x}$sinxx​	$0.99833$0.99833	$0.99998$0.99998	$0.99999$0.99999		$0.99999$0.99999	$0.99998$0.99998	$0.99833$0.99833

Whether we approach from the left (negative side) or the right (positive side), the value of $\frac{\sin x}{x}$sinxx​ approaches $1$1. This is further confirmed by looking at the graph for the function:

We therefore write

$\lim_{x\rightarrow0}\frac{\sin x}{x}=1$limx→0sinxx​=1.

This does not say that $\frac{\sin0}{0}=1$sin00​=1!  The function is still not defined for $x=0$x=0. But we can say that the function value becomes as close as we like to $1$1 by approaching $0$0.

Sometimes limits towards a finite value of $x$x can produce infinities. Let's reconsider the graph of $\frac{1}{x^2}$1x2​:

In this example, as $x$x approaches $0$0, the value of the function becomes larger and larger. Let's move the function over a little to see this better; here is the graph of $\frac{1}{\left(x-1\right)^2}$1(x−1)2​:

Here the function value becomes larger and larger as  $x$x approaches $1$1. The dashed line is called a vertical asymptote. Just like the horizontal asymptote we saw earlier, this is a line that the function draws arbitrarily close to. In limit form we write

$\lim_{x\rightarrow0}\frac{1}{x^2}=\infty$limx→01x2​=∞, and $\lim_{x\rightarrow1}\frac{1}{(x-1)^2}=\infty$limx→11(x−1)2​=∞.

There are a few things to keep in mind:

1. A function with an undefined value doesn't automatically have an asymptote. The function $f\left(x\right)=\frac{\sin x}{x}$f(x)=sinxx​ does  not have a vertical asymptote at $x=0$x=0, since the function doesn't look like a vertical line there.
2. A function can have an asymptote without having a limit, which occurs if the path of approach changes the behavior of the function near the limiting value.

To illustrate the second point, consider the graph of $y=\frac{1}{x-1}$y=1x−1​ as $x$x approaches $1$1:

Notice how approaching the limiting value from the left produces numbers that are larger and more  negative, while approaching the limiting value from the right produces numbers that are larger and positive. The function has a vertical asymptote at $x=1$x=1, but

the limit $\lim_{x\rightarrow1}\frac{1}{x-1}$limx→11x−1​ does not exist.

A similar thing is true for the function $f\left(x\right)=\tan x$f(x)=tanx. It has infinitely many $x$x-values for which the function is undefined, and infinitely many asymptotes, though the limit as $x$x approaches any of these values does not exist:

The function moves in opposite directions on either side of an asymptote.

 

Each of these concepts will be explored in greater detail in the coming lessons. For now, let's summarize what we've seen in this chapter.

 

Practice questions

Question 1

Question 2

 

Find limits from a graph or table of values

Although we will need a rigorous method of determining that a limit exists at a given point in the domain of a function, and what its value is, we can often resolve this issue confidently by looking at a graph or by constructing a table of values for points close to the point in question.

If a function reaches arbitrarily large positive or negative values as some value of the variable is approached, then no limit exists at that point. The function is said to have a vertical asymptote at that point. The function is undefined there and is said to be discontinuous at that point.

 

 

If in crossing some point in the domain, the function values jump without passing through intermediate values, we say the function has a jump discontinuity at that point. The limit as the point is approached from below will be different from the limit as the point is approached from above.

We may speak of a left-hand and a right-hand limit in this case but a strict limit does not exist at that point. (The notations $\lim_{x\rightarrow a^-}$limx→a− and $\lim_{x\rightarrow a^+}$limx→a+ are used for these left- and right-hand limits.)

 

A function can fail to be continuous at some point but still have a limit there. The following graph illustrates such a case. The function value need not be the same as the value of the limit at a given point.

 

 

Graphs and tables may be sufficient to allow the conclusion that a limit does not exist at a given point but to be certain that a limit does exist, more care is needed. The thinking is essentially the following:

Suppose a number $L$L is thought to be the limiting function value as a certain point $a$a in the domain is approached. To be certain that the limit exists, we require that the function attains values close to $L$L when the variable is sufficiently close to $a$a. This property has to hold however small we choose the distance of a function value to be from $L$L.

If the function has values arbitrarily close to $L$L for domain values sufficiently close to $a$a, then we can conclude that the limit as $a$a is approached is $L$L.

 

Roughly speaking, if the graph of a function sufficiently close to but surrounding a point looks like a straight line graph, then the function has a limit at that point and it is equal to the function value if it exists.

 

Consider the graph below, of the quadratic function defined on the interval $(-3,3)$(−3,3) given by $y=\frac{x^2}{2}-x+\frac{1}{2}$y=x22​−x+12​.

At $x=2$x=2, the function value is $\frac{1}{2}$12​. It is evident from the graph that if the value $2$2 is approached from below or from above, the corresponding function values approach $\frac{1}{2}$12​. We conclude that $\lim_{x\rightarrow2}=\frac{1}{2}$limx→2=12​.

Similar limit statements can be made about every point inside the domain interval. In each case, the limit is equal to the function value.

 

Practice questions

question 3

question 4