# Math Help - One Sided Limit Problem

1. ## One Sided Limit Problem

I need to find the lim as x approaches 9 from the left of this function:

f(x) = ((sqrt(x))-3)/(x-9)

I know from creating a table of y values when x is near 9 and from looking at its graph that the limit from either side is about 1.6666... , but I have been trying to manipulate the function to get in into a form that allows me to put 9 into the equation and actually solve for an exact answer. I can seem to figure out how. Any ideas?

2. ## Re: One Sided Limit Problem

Hey mattjp213.

You might want to check if you can use l'Hopitals rule here given the nature of your limit (it should).

3. ## Re: One Sided Limit Problem

Hint: $(\sqrt{x}+3) \text{ is a factor of } (x-9)$ and so is (...)

You'll get $\lim_{x \to 9}f(x) = \frac{1}{6}$ then you're done.

4. ## Re: One Sided Limit Problem

Hello, mattjp213!

$\displaystyle \lim_{x\to9^-}\frac{\sqrt{x}-3}{x-9}$

Multiply by $\frac{\sqrt{x} + 3}{\sqrt{x}-3}$

. . $\lim_{x\to9^-}\frac{\sqrt{x}-3}{x-9}\cdot\frac{\sqrt{x}+3}{\sqrt{x}+3} \;=\; \lim_{x\to9^-}\frac{x-9}{(x-9)(\sqrt{x}+3)}$

. . . $=\; \lim_{x\to9^-}\frac{1}{\sqrt{x} + 3} \;=\;\frac{1}{\sqrt{9}+3} \;=\;\frac{1}{3+3} \;=\;\frac{1}{6}$

5. ## Re: One Sided Limit Problem

Originally Posted by Soroban
Hello, mattjp213!

Multiply by $\frac{\sqrt{x} + 3}{\sqrt{x}-3}$

. . $\lim_{x\to9^-}\frac{\sqrt{x}-3}{x-9}\cdot\frac{\sqrt{x}+3}{\sqrt{x}+3} \;=\; \lim_{x\to9^-}\frac{x-9}{(x-9)(\sqrt{x}+3)}$

. . . $=\; \lim_{x\to9^-}\frac{1}{\sqrt{x} + 3} \;=\;\frac{1}{\sqrt{9}+3} \;=\;\frac{1}{3+3} \;=\;\frac{1}{6}$

Thanks for giving him the exact solution so he doesn't have to think about it.