1. ## another basic limit

How do I evaluate without using L'Hopital's rule
lim sqrt(x^2-9)/abs(3-x) as x->3-

I didn't manage to manipulate the algebric expression. I got to sqrt((x+3)/(x-3)) as x->3- and I don't know what to do from here

2. Originally Posted by GIPC
How do I evaluate without using L'Hopital's rule
lim sqrt(x^2-9)/abs(3-x) as x->3-

I didn't manage to manipulate the algebric expression. I got to sqrt((x+3)/(x-3)) as x->3- and I don't know what to do from here
Since you are approaching $\displaystyle 3$ from the left, you need to take notice of what happens with the values of $\displaystyle x < 3$.

Notice that $\displaystyle |x - 3| = -(x - 3) = 3 - x$ if $\displaystyle x < 3$.

So $\displaystyle \lim_{x \to 3^{-}}\frac{\sqrt{x^2 - 9}}{|x - 3|} = \lim_{x \to 3^{-}} \frac{\sqrt{x^2 - 9}}{3 - x}$

$\displaystyle = \lim_{x \to 3^{-}}\frac{x^2 - 9}{(3 - x)\sqrt{x^2 - 9}}$

$\displaystyle = \lim_{x \to 3^{-}}\frac{-(3 - x)(x + 3)}{(3 - x)\sqrt{x^2 - 9}}$

$\displaystyle = \lim_{x \to 3^{-}}\frac{-(x + 3)}{\sqrt{x^2 - 9}}$.

This now tends to $\displaystyle -\frac{6}{0}$ which tends to $\displaystyle -\infty$.

3. and how did you conclude that 6/0 tends to infinity? I can't use L'Hopital's rule in this one, how can I explain this result otherwise?

4. Originally Posted by GIPC
and how did you conclude that 6/0 tends to infinity? I can't use L'Hopital's rule in this one, how can I explain this result otherwise?
How many 0's go into 6?

0 + 0 + 0 + ... = 0

So infinitely many will go into 6.

5. I understand the intuitive explanation but if I had to submit the homework, How do I show it on a more formal note?

6. Originally Posted by GIPC
I understand the intuitive explanation but if I had to submit the homework, How do I show it on a more formal note?
It's well known that $\displaystyle \lim_{0 \to \infty}\frac{a}{x} = \infty$ for $\displaystyle a > 0$.

Check the graph of $\displaystyle \frac{1}{x}$.

It's one of the most basic limits and one you will need to remember.

7. Originally Posted by Prove It
It's well known that $\displaystyle \lim_{x \to \infty}\frac{a}{x} = \infty$ for $\displaystyle a > 0$.

Check the graph of $\displaystyle \frac{1}{x}$.

It's one of the most basic limits and one you will need to remember.
Typo- that should be $\displaystyle \lim_{x\to 0}\frac{a}{x}= \infty$.

8. Originally Posted by HallsofIvy
Typo- that should be $\displaystyle \lim_{x\to 0}\frac{a}{x}= \infty$.
Thanks, will edit now.