1. ## Limits

I'm learning about limits and I have realised that if you graph the limit function, it seems to never exist where x = a i.e. where x = the number you're trying to calculate the limit at. However in the same tutorial it mentions that "many of the functions don't exist at x= a". This to me seems wrong, to me it seems functions never exist at x = a. Am I correct?

2. ## Re: Limits

No. $$\lim_{x \to 4} (x-2) = 2$$

Indeed, the definition of the continuity of a function at a given point is that the function is equal to the limit of the function.

3. ## Re: Limits

It depends on the function whose limit you are analyzing and whether that function is actually defined at x = a. For example:

Suppose you are given $\displaystyle f(x)=x^2$

$\displaystyle \lim_{x\to1}\left(f(x)\right)=1$

This comes directly from the fact that $\displaystyle f(1)=1$. Now consider:

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

Here, we would find:

$\displaystyle \lim_{x\to1}\left(g(x)\right)=0$

Even though $\displaystyle g(1)$ is undefined.

4. ## Re: Limits

The examples you are given have that property because they are more interesting. But continuous functions which, by definition, have the property that $\displaystyle \lim_{x\to a} f(x)= f(a)$ are the most useful functions.

5. ## Re: Limits

I think where my confusion arose from was the fact that I was introduced to limits through derivatives. So I thought all limits were of the derivative form. Looks like the derivative is just a special case of limits.

Yes.