limit (x approaches 13) (x - 13) / (sqrt ( x + 3 ) - 4)

[annajee]

I am trying to figure out this problem:

lim (x approaches 13) (x - 13) / (sqrt ( x + 3 ) - 4)

My calculator says error, another calculus calculator says the limit does not exist, but my solution key to this problem says the limit is 8.

It says to multiply both the numerator and the denominator by (sqrt (x + 3) + 4). and that somehow then the limit is 8. But, how does it figure its 8? It would still be 0 / 0. Right?

Can someone explain this to me, how that kind of multiplying would make the limit 8?

Thanks
Anna

 

[stapel]

...my solution key to this problem says...to multiply both the numerator and the denominator by (sqrt (x + 3) + 4)....

What did you get when you followed the instructions? (I can't imagine how you arrived at (sqrt[x + 3] - 4)(sqrt[x + 3] + 4) equalling zero...?)

Please be complete. Thank you!

Eliz.

 

[galactus]

Try this:

\(\displaystyle \lim_{x\to 13}\frac{(x-13)}{(\sqrt{x+13}-4)}\cdot\frac{(\sqrt{x+3}+4)}{(\sqrt{x+3}+4)}\)

Factor:

\(\displaystyle \lim_{x\to 13}\frac{\sqrt{x+3}(x-13)+4(x-13)}{x-13}\)

\(\displaystyle \lim_{x\to 13}\frac{(\sqrt{x+3}+4)(x-13)}{x-13}\)

See it now?.