## anonymous one year ago Limit of x-3/((root of x+6)-3) as x approaches 3?

1. freckles

try multiply the conjugate of the denominator on top and bottom

2. anonymous

would the conjugate be $\sqrt{x+6}+3$ ?

3. freckles

yep!

4. anonymous

thank you!

5. freckles

so you already got the limit?

6. freckles

$\lim_{x \rightarrow 3}\frac{x-3}{\sqrt{x+6}-3} \\ \lim_{x \rightarrow 3}\frac{(x-3)(\sqrt{x+6}+3)}{(\sqrt{x+6}-3)( \sqrt{x+6}+3)}$ and you should see: $(\sqrt{x+6}-3) (\sqrt{x+6}+3)=(x+6)-9=x-3$ so a common factor from top and bottom will "cancel"

7. anonymous

And then the answer would be 6?

8. freckles

sqrt(3+6)+3 sqrt(9)+3 3+3 yes 6

9. anonymous

Thank you!

10. freckles

np

11. idku

$\lim_{x \rightarrow 3}\dfrac{x-3}{\sqrt{x+6}-3}$when you plug in x=3, you get 0/0, thus you can differentiate on top and bottom. $\lim_{x \rightarrow 3}{\Large \frac{1-0}{\frac{1}{2\sqrt{x+6}}-0}}$ $\lim_{x \rightarrow 3}{\Large \frac{1}{\dfrac{1}{2\sqrt{x+6}}}}$ $\lim_{x \rightarrow 3}2{\sqrt{x+6}}$$2{\sqrt{3+6}}=6$

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