Question

Create or find your own, unique function h(x). discuss the limit of your function as latex: x x approaches some particular value, let's say a, where the limit exists but the y-value does not or is different than the limit.

Answer 1

Consider the function:

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

As x tends to 1, 

$h(1)= \frac{1^2-1}{1-1} = \frac{0}{0}$

hence, the y value does not exist.

But

$\lim_{x\to1} h(x)= \lim_{x\to1} \frac{x^2-1}{x-1} \\ \\ = \lim_{x\to1} \frac{(x-1)(x+1)}{x-1}= \lim_{x\to1} (x+1) \\ \\ =1+1=2$

Therefore, the limit of h(x) exists but the y-value does not exist.