Calculus/Finite Limits

Informal Finite Limits

The first way we can approach the problem is to say that at $x=1.99$, $f(x)=3.9601$. While this is pretty close, the function may do something completely different later on. For instance, let's pick the function $f(x)=x^4-2x^2-3.77$; at 1.99 it's even closer to 4 than 3.9601, but at 2 it's almost four and a quarter. It seems like maybe we should go even closer. The problem is, no matter how close we get, it will not be close enough for some functions, and we won't know what they do at that point.

The solution is to find out what happens arbitrarily close to the point. This means that no matter how close we want the function to get to 4, we can find an x close enough to 2 such that it will get there. In our example, when x gets really close to 2, f(x) should get really close to 4. We will write this as

$\underset{x\to 2}{\lim }f(x)=4$

which is said, "The limit, as x approaches 2, equals 4," or, "As x approaches 2, f(x) approaches 4."

The advantage of this approach is that we can quickly see that, no matter how close we make x to 2, it doesn't make $x^4-2x^2-3.77$ really close to 4. In fact, if we want $3.99<f(x)<4.01$, x has to stay pretty far away from 2, relatively speaking.

Plot of x*sin(1/x) for -0.5 < x <0.5

This idea of talking about a function as it approaches something was a major breakthrough, because it let us understand things that we could previously only approximate. Let's say, for instance, that we want to find how the function $x\sin (1/x)$ works really close to 0. The graph of this function "wiggles" too much to let us see what is going on, and $x\sin (1/x)$ does not exist at 0, but using limits, we can show that the function is, in fact, going to be arbitrarily close to 0; the limit is 0.

One-Sided Limits

Sometimes, it is necessary to consider what happens when we approach an x value from one particular direction. To accommodate 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 $\underset{x\to 2}{\lim }\sqrt{x-2}$, there is a problem because there is no way for x to approach 2 from the left hand side (the function is undefined here).

The left-handed limit: $\underset{x\to 2^{-}}{\lim }\sqrt{x-2}=\mathrm{\varnothing }$

The right-handed limit:$\underset{x\to 2^{+}}{\lim }\sqrt{x-2}=0$

Since the limits are different from each side, we have: $\underset{x\to 2}{\lim }\sqrt{x-2}=\mathrm{\varnothing }$