# Thread: Limit exist or not?

1. ## Limit exist or not?

Find the following limit, if it exist:
$\displaystyle \lim_{x \to 1}\frac{x + 2}{x^2 + x - 2}$

apply L'Hospital's Rule,
i get, $\displaystyle \frac{1}{3}$

so the limit is exist..
correct?

2. Originally Posted by nameck
Find the following limit, if it exist:
$\displaystyle \lim_{x \to 1}\frac{x + 2}{x^2 + x - 2}$

apply L'Hospital's Rule,
i get, $\displaystyle \frac{1}{3}$

so the limit is exist..
correct?
You can't apply L'Hospital's rule, because it does not tend to $\displaystyle \frac{0}{0}$ or $\displaystyle \frac{\infty}{\infty}$.

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

$\displaystyle = \frac{1}{x - 1}$.

So $\displaystyle \lim_{x \to 1}\frac{x + 2}{x^2 + x - 2} = \lim_{x \to 1}\frac{1}{x - 1}$

Try graphing this function. I think you'll see that the left hand limit is $\displaystyle -\infty$ and the right hand limit is $\displaystyle \infty$.

Since the left and right hand limits don't match, the limit does not exist.

3. Thanks Prove It!