Expert: Paul Klarreich - 3/3/2008

Question
HOW TO SOLVE lim x tends to infinity { [3+lnx]/[2lnx-1]}

Answer
Questioner:   krishnamoorthy
Category:  Calculus
Private:  No
 
Subject:  CALCULUS INDETERMINATE FORM
Question:  HOW TO SOLVE lim x tends to infinity { [3+lnx]/[2lnx-1]}
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Hi, Krishnamoorthy,

[My bridge partner is Vaidhyanathan; in some locales he is known as Vidi, in others as Nathan.  What do you prefer?]

Your limit as  x --> infinity, looks pretty straightforward.

Let  y = ln x; then as  x->infinity, so does y. Now we can write:

 3+lnx
----------- =
2 lnx - 1

 3+y
------- = (after dividing every term by y]
2y - 1


3/y + 1
-------
2 - 1/y

Now, when y -> infinity,  3/y and 1/y both approach zero.  You have:

0 + 1
----- = 1/2
2 - 0