Expert: Paul Klarreich - 9/17/2007

Question
limit as x approaches 0 of x/sin3x. I want to figure it out without using L'hopitals rule. Is that possible?

Answer
Questioner:   Lauren
Category:  Calculus
Private:  No
 
Subject:  limits
Question:  limit as x approaches 0 of x/sin3x. I want to figure it out without using L'hopitals rule. Is that possible?
................................................
Hi, Lauren,

You will use this well-known limit:
     sin t  
lim   ----- = 1
t->0    t

Now take your example:

        x   
lim   --------
x->0   sin 3x

and substitute   t = 3x.  Then you deduce:

If  t = 3x,  x = t/3.

If  x -> 0, then  3x -> 0,  so  t -> 0, and you have:

       t/3
lim   -------
t->0   sin t

Now a bit of algebra:

        t
lim   ------- =
t->0  3 sin t

     1    t
lim   - ------ =
t->0  3  sin t

     1     1
lim   - -------- =
t->0  3  sin t/t

1            1
-  lim   -------- =
3  t->0   sin t/t

1         1
-  ------------------ =
3  lim [t->0] sin t/t

1   1     1
-  --- = ---
3   1     3