Extension: Define Two-Sided Limits in Terms of Continuity

Now that we understand continuity reasonably well, let's try to look at limits.

The idea of a "limit" essentially means "getting really close". For functions that are continuous at a point, the limit at that point is very easy: it's the same as the function's actual value.

If that's all there were to it, the idea of a limit would be really boring. But we can study the limit of certain functions that aren't continuous at a point. Particularly, they can have a limit when they're not defined.

The value of a two-sided limit is just the value the function would have at that point in order to be continuous.

In particular, the limit does not depend on what the function's value actually is at that point.

Looking at the graph may make it easier to understand. If we ignore the dot at $x = 2$ itself, which is how limits work, we can observe that the function gets really close to $4$ when $x = 2$.

You may complain that this function is also boring --- why not just simplify it to $x + 2$? But some functions don't allow that.

The graph looks like this:

This function is more interesting because, even though we can't easily simplify it to avoid the division by zero, there's a value we can define it to have at zero that makes it continuous. That's the limit!

Of course, not all functions have limits.

If we look at the plot again, we can maybe see why. The function becomes infinitely large. No value we could assign to the function at $0$ could make it continuous, because the function would just go right past it.

Here's another example.

The plot looks like this:

Here, even though the function stays reasonably small, there's a "gap" between the left part and the right part that a single value couldn't possibly bridge to make it continuous, Therefore, the limit at 0 is not defined.

(However, the limits on one side are defined. We'll look at this more later.)

Having gone through all this, it's time to try writing down a definition for the two-sided limit.

It's actually very similar to the definition for continuity. Try writing down a definition.

We can replace a few expressions in our definition of continuity to create this:

Definition? The limit of a function $f(x)$ at $a$ is a value $b$ such that, for all $\epsilon > 0$, we can find a number $\delta$ such that $f(x)$ is within $\epsilon$ of $b$ for all values of $x$ within $\delta$ of $a$.

But this definition misses one of the most important parts of the limit. Can you see what it is?

This definition actually involves the value of $f(a)$ itself, because $a$ is certainly within $\delta$ of $a$ for any $\delta > 0$. So we have to patch that as follows:

Definition: The limit of a function $f(x)$ at $a$ is a value $b$ such that, for all $\epsilon > 0$, we can find a number $\delta$ such that $f(x)$ is within $\epsilon$ of $b$ for all values of $x$ within $\delta$ of $a$ but not equal to $a$ itself.

That is the definition of the two-sided limit. It is also written $\lim_{x\to a} f(x) = b$.

The limit of a function exists if and only if the function is continuous at that point.

When is a function NOT continous?

1. If it has a hole at the point

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1. When the left hand limit does not equal the right hand limit

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1. When the function has asymptotes

Note: If you unsure whether a function is continous or not think about whether you can draw the function's graph without lifting your pen or pencil off the paper. If you can do that the function is continuous.