another basic limit

**GIPC** · April 13th 2010, 08:53 PM

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

**Prove It** · April 13th 2010, 09:47 PM

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 $3$ from the left, you need to take notice of what happens with the values of $x < 3$ .

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

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

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

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

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

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

**GIPC** · April 13th 2010, 11:28 PM

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?

**Prove It** · April 13th 2010, 11:31 PM

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.

**GIPC** · April 14th 2010, 12:26 AM

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

**Prove It** · April 14th 2010, 12:45 AM

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 $\lim_{0 \to \infty}\frac{a}{x} = \infty$ for $a > 0$ .

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

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

**HallsofIvy** · April 14th 2010, 02:37 AM

Originally Posted by Prove It

It's well known that $\lim_{x \to \infty}\frac{a}{x} = \infty$ for $a > 0$ .

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

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

Typo- that should be $\lim_{x\to 0}\frac{a}{x}= \infty$ .

**Prove It** · April 14th 2010, 02:55 AM

Originally Posted by HallsofIvy

Typo- that should be $\lim_{x\to 0}\frac{a}{x}= \infty$ .

Thanks, will edit now.

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