Calculus/Limits and their Properties/Contents

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Limits and their Properties

In this chapter, you will study limits, which will allow you to get an idea of the behavior of a function at or near every point, regardless of whether or not the function is undefined at that particular point. You will understand the limit of the function as it approaches that point.

An intuitive look at limits

Let's start our look at limits with a little mind experiment.

Let's Imagine a flea on a table. This particular flea will do exactly half of whatever you tell him to do. If you place him at one end of the table and tell him to jump to the other side he will jump exactly half way across. Tell him to jump again and he will end up a quarter of the way from the edge. What do you think will happen if you keep telling the flea to jump to the edge forever? Will he ever make it to exactly the edge? If he doesn't, how far will the flea travel in total?

Let's try another idea. Imagine you are measuring the height of a cave. This particular cave has an infinitely big opening at the mouth. When you enter, however, the caves ceiling gets lower and lower. As you get deeper into the cave you start to realize the ceiling is desending in proportion to the distance you walk, such that when you were 1 foot into the cave the ceiling is 100 feet high, at 10 feet the ceiling is 10 feet high and 100 feet down the cave the ceiling is 1 foot off the ground. If fact, lets say the cave's height equals y and y=100/x where x is the cave's depth

A cross section of this cave would probably look something like this:

As you can see, the cave gets smaller and smaller as you crawl deeper down the passage.

Now what do you think would happen deep into the cave? What would be the height of the cave toward the far end, if the cave streched out to infinity? Would the ceiling ever touch the floor?

Both of these experiments are situations where limits can be useful. While we could never get an exact measure of how far the flea would travel in infinite jumps, since he always getting halfway closer to the edge, his total distance would have a limit equal to the total length of the table. The flea jumps and jumps and gets closer and closer but never quite there. Likewise, if you continued to walk down the cave forever, (and you were very very short), the cave would get very small but would never touch the floor. The height of the cave at an infinite distance into the cave would have a limit equal to zero. It would get very close to zero but never quite there.

Terminology

Let f(x) be a function of x. If we want to talk about how f(x) behaves as x approaches a certain value, then we write $\lim_{x \to n} f (x) = y$ This means that as x gets closer and closer to n, f(x) gets closer and closer to y.

Sometimes we want to study the behavior of f(x) as x approaches a certain value from just the left side or right side. In that case, we write $\lim_{x \to n +} f (x) = y$ to indicate the right-side limit, and $\lim_{x \to n -} f (x) = y$ to indiate the left-side limit.

Note that we use "approaches" very loosely and (hopefully) intuitively here. More precise definitions are available and can be found in textbooks on Real Analysis. For our purposes, only a basic understanding of limits is required. When we say x approaches n, or x gets near n, we mean that x takes on values that are very very close to n, but not n exactly. When we're talking about regular limits, these values of x can be on both sides of n. For instance, if n = 3, and x approaches n, you can think of x as 2.9999999999 and 3.00000000001. When we're talking about left-or-right-side limits, then x is restricted to that particular side of n.

Simple Limits

The simplest type of limit is one where you can simply substitute the value you are approaching into the function and get a valid result. For example, let $f (x) = x^{2}$ . Then $\lim_{x \to 3} x^{2} = 3^{2} = 9$ .

Looking at the graph of $y = x^{2}$ , you can see that as x gets nearer and nearer to 3, y gets nearer and nearer to 9, confirming our calculation. (insert graph of y = x^2 here)

Example 1

Find $\lim_{x \to 0} (x + 3)^{2} + 2 x - 1$ .

Solution: We substitute x = 0 into the formula to get $(0 + 3)^{2} + 2 (0) - 1 = 3^{2} - 1 = 8$ . Thus, the limit is 8.

Limits at Holes

A more complicated type of limit involves finding the value of a function as it nears a "hole", or a place where its value is not defined. For instance, consider $f (x) = \frac{x^{2} - 9}{x - 3}$ . In the graph of $y = f (x)$ , there is a hole at x = 3 because substituting that value into the equation produces division by 0.

What about $\lim_{x \to 3} f (x)$ ? From the graph it is clear that as x approaches 3, y approaches 6.

(insert graph here)

So, even though the function is undefined at x = 3, the limit is defined. To see why, watch what happens when we simplify f(x) using regular algebra:

$f (x)$	$= \frac{x^{2} - 9}{x - 3}$
	$= \frac{(x + 3) (x - 3)}{x - 3}$
	$= x + 3$

After simplifying, we can see that $\frac{x^{2} - 9}{x - 3}$ is the same as $x + 3$ , except at the hole, so it's little surprise that $\lim_{x \to 3} f (x) = 6$ . However, always remember that the original function is still undefined at x=3! The function before simplification is not the same as the function after simplification. If we were using this equation to model joint strengths of a bridge, then perhaps x = 3 represents some condition that causes the bridge to fail. Just because we can cancel that out algebraically does not mean the bridge will hold up.

Example 2

Find $\lim_{x \to 1} \frac{x^{2} - 2 x + 1}{x - 1}$ .

Since substituting x = 1 into the function produces a division by 0, we try to simplify and cancel whatever we can.

$f (x)$	$= \frac{x^{2} - 2 x + 1}{x - 1}$
	$= \frac{(x - 1) (x - 1)}{x - 1}$
	$= x - 1$

Now we substitute x = 1 into this new function and we get 0. Thus, the limit is 0.

When Left and Right Don't Meet

So far, we've looked at the limits of simple functions that can be found by substitution and the limits of more complex functions that can be found by simplification. Now it is time to look at some cases where limits are undefined.

Consider the function $f (x) = \frac{x}{| x |}$ and study the behavior of f(x) as x nears 0. When x is positive, we know that $| x | = x$ so $f (x) = \frac{x}{| x |} = \frac{x}{x} = 1$ . Thus, for very small positive x values, like 0.00000001, f(x) = 1. When x is negative, on the other hand, $f (x) = \frac{x}{| x |} = \frac{x}{- x} = - 1$ . Thus, for very small negative values, like -0.000000001, f(x) = -1. What, then, is $\lim_{x \to 0} f (x)$ ? We know that we have to consider values very close to 0, on both the left and the right. But the function changes very abruptly from -1 to 1 in that region. In this case, we say that the limit doesn't exist.

Interestingly, $\lim_{x \to 0 +} f (x)$ and $\lim_{x \to 0 -} f (x)$ do exist, with values of 1 and -1 respectively. Even though the function's limit exists on the left of 0 and the right of 0, the limit values don't "meet up" properly. We can generalize this by saying that for a limit to exist, we require that

$\lim_{x \to n +} f (x) = \lim_{x \to n -} f (x)$

And we say the limit does not exist if

$\lim_{x \to n +} f (x) \neq \lim_{x \to n -} f (x)$