Expert: Paul Klarreich - 12/1/2007

Question
Hello,

I am needing help with another of my homework problems. Can you please tell me how I should go about solving the following problem?

limit as x->2 ((2/ln(x-1)) - (2/(x-2)))

Thank you,

Danielle

Answer
Questioner:   Danielle
Category:  Calculus
Private:  No
 
Subject:  Indeterminate limits
Question:  Hello,

I am needing help with another of my homework problems. Can you please tell me how I should go about solving the following problem?

limit as x->2 ((2/ln(x-1)) - (2/(x-2)))

Thank you,

Danielle
----------------------------------------------
        2        2  
lim   ------- - ------
x->2  ln(x-1)   x - 2

Have you studied l'Hospital's rule?  I hope so, because that seems the way to go.
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WARNING: USE COURIER FONT TO VIEW THIS
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But, I hear you say, that only applies to one-term stuff like:

inf     0
--- or ----
inf     0

and this is  inf - inf.  The answer is, if you MUST have a single term, then MAKE ONE.[The Clint Eastwood Principle: A man's gotta do what a man's gotta do.]

Combine fractions:  LCD = ln(..)(x - 2)

  2(x - 2) - 2ln(x - 1)
  ---------------------
  ln(x-1)(x - 2)

Now you have  0/0 and we can do it.  Diff the top:
                                         
D[2(x - 2) - 2ln(x - 1)] = 2 - 2/(x-1)  
 2x - 2 - 2                                       
= -----------     
    x - 1

 2x - 4  
= -------
  x - 1

Now the bottom:(product rule)

                                    
D[ln(x-1)(x - 2)] =  
               x - 2
ln(x - 1)(1) + -------
               x - 1

 (x - 1)ln(x - 1) + x - 2
= ------------------------
        x - 1

Whew! Now put it together and the x-1's in the denominators cancel.
      2x - 4  
--------------------------
 (x - 1)ln(x - 1) + x - 2

Now try your  x -> 2 again:

Oops! another 0/0.  What can we do?  Repeat the process:

D[2x - 4] = 2   << TOP


D[(x - 1)ln(x - 1) + x - 2] =  

x - 1
------ + 1 ln(x - 1) + 1 =
x - 1

1 + ln(x - 1) + 1 =

2 + ln(x - 1)   << BOTTOM

Here we go again.  The fraction is:

       2
---------------
2 + ln(x - 1)

AND FINALLY, x ->2 gives  2/2 = 1