Expert: Paul Klarreich - 4/14/2007

Question
if {f(x)=(sqrt(2x+5)-sqrt(x+7))/(X-2), for x
not equal to 2 and f(2)=k and if f is continuous at x=2, then what does
k=?  

Answer
Questioner:   tina
Category:  Calculus
 
Subject:  calculus
Question:  if {f(x)=(sqrt(2x+5)-sqrt(x+7))/(X-2), for x
not equal to 2 and f(2)=k and if f is continuous at x=2, then what does
k=?
.........................
Hi, Tina,

You have this function, clearly undefined at  x = 2 because of that nasty  x-2  in the denominator:

      sqrt(2x+5)-sqrt(x+7)
f(x) = ---------------------
             x - 2

You need the limit of f(x) as x -> 2.  For this, you ABSOLUTELY, POSITIVELY MUST get that factor of  x-2  out of there.  

Basic clue:  you see two terms involving square roots, RATIONALIZE.

WARNING: THIS DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL DIFFICULT TO VIEW ON CERTAIN COMPUTING SYSTEMS.  VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.


      (sqrt(2x+5)-sqrt(x+7))  (sqrt(2x+5)+sqrt(x+7))
f(x) = ----------------------- ----------------------
             x - 2            (sqrt(2x+5)+sqrt(x+7))

          2x+5 - (x+7)
f(x) = -------------------------------
       (x - 2) (sqrt(2x+5)+sqrt(x+7))


            2x+5 - x-7
f(x) = -------------------------------
       (x - 2) (sqrt(2x+5)+sqrt(x+7))


            x - 2
f(x) = -------------------------------
       (x - 2) (sqrt(2x+5)+sqrt(x+7))

             1
f(x) = ----------------------
      sqrt(2x+5)+sqrt(x+7)

NOW you can put  x = 2 without any trouble.

             1
f(2) = -----------------------
      sqrt(2(2)+5)+sqrt(2+7)


             1
f(2) = ---------------
      sqrt(9)+sqrt(9)

        1      1
f(2) = ----- = ---
      3 + 3    6
That's it.