lim h-- 0

((Sqrt 9+h)-3)/h

Question

lim h--> 0

((Sqrt 9+h)-3)/h

tkhunny · Answer

"Rationalize" the Numerator. -- This may be contrary to anything you have ever done befire.

anonymous · Answer

Alrightt...So would I multiply the top and bottom by Sqrt 9-h?

tkhunny · Answer

Not quite.  Try $\sqrt{9+h} + 3$

anonymous · Answer

Hmm...I did that and I got h(h+6) for the numerator and (Sqrt(9+h)+3)h And I had the h's cancel then substituted the zero and got 6/6 or 1 However when I enter it it says it's wrong

tkhunny · Answer

Something funny in that numerator...

$(\sqrt{9+h}-3)(\sqrt{9+h}+3) = (9+h) - 9 = h$

anonymous · Answer

Use your knowledge of "completing the square" when you're simplifying the numerator.

anonymous · Answer

Ahhh! I finally got it! Thanks a lott! Ha :)

whpalmer4 · Answer

$$\frac{\sqrt{9+h} - 3}{h}$$I like to make life a little easier with these sorts of things by substituting another letter for the radical expression: $a = \sqrt{9+h}$
Now we have $$\frac{a-3}{h}*\frac{a+3}{a+3} =\frac{a^2-9}{h(a+3)}$$and now we substitute back to get
$$\frac{9+h-9}{h(\sqrt{9+h}+3)} = \frac{1}{\sqrt{9+h}+3}$$ and it is easy to evaluate that at $h=0$.  Not a big gain in this problem, but if we were working with more complicated expressions, only using the radicals at the end can save some effort, spotting factors may be easier, etc.  YMMV.