Calculus/Finite Limits

From Wikibooks, the open-content textbooks collection

← Limits	Calculus	Infinite Limits →
	Finite Limits

[edit] Informal Finite Limits

Now, we will try to restate more carefully the ideas of the last chapter. We said then that the equation $\lim_{x\to 2} f(x) = 4$ meant that, when $x$ gets close to 2, $f (x)$ gets close to 4. But what does this mean, exactly? How close is "close"? The first way we can approach the problem is to say that, at $x = 1.99$ , $f (x) = 3.9601$ , say, which is pretty close to 4. But the function may do something completely different later on. For instance, suppose $f (x) = x 4 - 2 x 2 - 3.77$ ; $f (1.99) = 3.99219201$ , which is even closer to 4 than 3.9601, but $f(1.999)=4.2060\dots$ --- more than four and a fifth. It seems like maybe we should go even closer than 1.99 is to 2. The problem is, no matter how close we get, it will not be close enough for some functions, so we can never be sure what they do even closer in.

The solution is to find out what happens arbitrarily close to the point. In particular, we want to say that, no matter how close we want the function to get to 4, if we make $x$ close enough to 2 then it will get there. In this case, we will write

$\quad\lim_{x\to 2} f(x) = 4$

and say "The limit of $f (x)$ , as $x$ approaches 2, equals 4" or "As $x$ approaches 2, $f (x)$ approaches 4." In general:

Definition: (New definition of a limit)

We call $L$ the limit of $f (x)$ as $x$ approaches $c$ if $f (x)$ becomes arbitrarily close to $L$ whenever $x$ is sufficiently close (and not equal) to $c$ .

When this holds we write

$\lim_{x \to c} f(x) = L$

or

$f(x) \to L \quad \mbox{as} \quad x \to c.$

The advantage of this approach is that we can quickly see that, no matter how close we make $x$ to 2, it doesn't guarantee that $x 4 - 2 x 2 - 3.77$ will be really close to 4. In fact, if we even just want to guarantee $3.9 < f (x) < 4.1$ , $x$ has to be less than 1.995.

[edit] One-Sided Limits

Sometimes, it is necessary to consider what happens when we approach an $x$ value from one particular direction. To account for this, we have one-sided limits. In a left-handed limit, $x$ approaches $a$ from the left-hand side. Likewise, in a right-handed limit, $x$ approaches $a$ from the right-hand side.

For example, if we consider $\quad\lim_{x\to 2} \sqrt{x-2}$ , there is a problem because there is no way for $x$ to approach 2 from the left hand side (the function is undefined here). But, if $x$ approaches 2 only from the right-hand side, we want to say that $\sqrt{x-2}$ approaches 0.

Definition: (Informal definition of a one-sided limit)

We call $L$ the limit of $f (x)$ as $x$ approaches $c$ from the right if $f (x)$ becomes arbitrarily close to $L$ whenever $x$ is sufficiently close to and greater than $c$ .

When this holds we write

$\lim_{x \to c^+} f(x) = L.$

Similarly, we call $L$ the limit of $f (x)$ as $x$ approaches $c$ from the left if $f (x)$ becomes arbitrarily close to $L$ whenever $x$ is sufficiently close to and less than $c$ .

When this holds we write

$\lim_{x \to c^-} f(x) = L.$

In our example, the left-handed limit $\quad\lim_{x\to 2^{-}} \sqrt{x-2}$ does not exist.

The right-handed limit, however, $\quad\lim_{x\to 2^{+}} \sqrt{x-2} = 0$ .

It is a fact that $\lim_{x\to c} f(x)$ exists if and only if $\lim_{x\to c^+} f(x)$ and $\lim_{x\to c^-} f(x)$ exist and are equal to each other. In this case, $\lim_{x\to c} f(x)$ will be equal to the same number.