Expert: Paul Klarreich - 4/19/2010

Question
I asked this question to someone else, but the answer they gave was incorrect. Here is the question:
I am generally pretty good with evaluating and simplifying expressions, but this one is a stumper:
Limit as x approaches 4
(sqrt(8-x)-2)/(sqrt(13-x)-3)
Any suggestions?
I know that the answer is 3/2 (using a calculator that can evaluate the limit)
The other responder recommended that I multiply by the reciprocal of the denominator, but doing so makes the denominator divide by zero when you evaluate it at that limit. The numerator does the same thing.
Can you recommend another way to evaluate it?
Thanks!

Answer
Questioner:   Jesse
Category:  Calculus
Private:  No
 
Subject:  evaluate the limit
Question:  I asked this question to someone else, but the answer they gave was incorrect. Here is the question: I am generally pretty good with evaluating and simplifying expressions, but this one is a stumper: Limit as x approaches 4 (sqrt(8-x)-2)/(sqrt(13-x)-3) Any suggestions? I know that the answer is 3/2 (using a calculator that can evaluate the limit) The other responder recommended that I multiply by the reciprocal of the denominator, but doing so makes the denominator divide by zero when you evaluate it at that limit. The numerator does the same thing. Can you recommend another way to evaluate it? Thanks!
         sqrt(8-x)-2)
lim[x->4] -------------
         sqrt(13-x)-3

Since this becomes  0/0, there should not be any trouble.

I suggest a 'double rationalization':

         sqrt(8-x)-2) * (sqrt(8-x) PLUS 2)/(sqrt(13-x) PLUS 3)
lim[x->4] ------------- ---------------------------------------
         sqrt(13-x)-3   (sqrt(8-x) PLUS 2)/(sqrt(13-x) PLUS 3)


         (8 - x) - 4   (sqrt(13-x) PLUS 3)
lim[x->4] ------------ ----------------------
         (13 - x) - 9    (sqrt(8-x) PLUS 2)


         4 - x   (sqrt(13-x) PLUS 3)
lim[x->4] ------ ----------------------
         4 - x    (sqrt(8-x) PLUS 2)

Cancel:

         (sqrt(13-x) PLUS 3)
lim[x->4] ----------------------
          (sqrt(8-x) PLUS 2)

and the rest is easy.