Expert: Paul Klarreich - 11/20/2008

Question
Would you please help me with following:

1)lim(x->9) ((x-10sqrt(x)+21)/(3-sqrt(x)))

I don't know how to even get started with this..

2)lim x->2 ((x^4+x^3-24)/(x^2-4))
=lim x->2 ((x-2)(x^3+3x^2+12)/((x-2)(x+2)))
=11 Is this right?

Answer
Questioner:   Linda
Category:  Calculus
Private:  No
 
Subject:  Limits
Question:  Would you please help me with following:

1)  lim(x->9) ((x-10sqrt(x)+21)/(3-sqrt(x)))

I don't know how to even get started with this..

Let's make it more legible:
          x - 10 sqrt(x) + 21
lim(x->9) ---------------------
               3 - sqrt(x)

First you check on substitution of x EQUALS 9:

  9 - 10 sqrt(9) + 21
---------------------
       3 - sqrt(9)

  9 - 30  + 21    0
-------------- = ----
     3 - 3          0

Bad news: zero on the bottom.  Good news: zero on top, so you are still in the game.

How about a nice rationalizing of the denominator?  (Think of that every time you see a square root.)


 x - 10 sqrt(x) + 21   3 + sqrt(x)
--------------------- ------------
      3 - sqrt(x)      3 + sqrt(x)  

(x - 10 sqrt(x) + 21)(3 + sqrt(x))
-----------------------------------
            9 - x

3x - 30 sqrt(x) + 63 + x sqrt(x) - 10x + 21sqrt(x)
---------------------------------------------------
            9 - x

 - 9 sqrt(x) + 63 + x sqrt(x) - 7x
-----------------------------------
            9 - x

Now the top BETTER HAVE A FACTOR OF  9-x, or .....

 (9 - x)(7 - sqrt(x))
---------------------  << yes, it does.
       9 - x

= 7 - sqrt(x)

Now put  x = 9,  you get 7 - 3 = 4.

END OF STORY.
.........................
2)
lim x->2 ((x^4+x^3-24)/(x^2-4))

=lim x->2 ((x-2)(x^3+3x^2+12)/((x-2)(x+2)))

=  11

Is this right?
 
11 is correct, but did you leave out a term?  You should have:

((x-2)(x^3+3x^2   -- PLUS 6x --  +12)/((x-2)(x+2)))

Then it's OK.