Homework Help: Limit of the form 0^infinity+infinity^0

1. Dec 8, 2009

abhineetK

1. The problem statement, all variables and given/known data
What is the value of
lim {(1/x)sinx+(sinx)1/x} ,x>0 ???
x->0

2. Relevant equations
??

3. The attempt at a solution
lim {(1/x)sinx+(sinx)1/x}
x->0
=lim (1/x)sinx+lim (sinx)1/x
..x->0............x->0

Let A=lim (1/x)sinx
.........x->0
and B=lim (sinx)1/x
.........x->0

logA=lim (sinx)log(1/x)=0
........x->0
=>A=e0=1

logB=lim (1/x)log(sinx)=infinity
........x->0
=>B=infinity

The problem is that the answer is 0.

Last edited: Dec 9, 2009
2. Dec 8, 2009

dextercioby

I understand that x goes to 0 in the limit, but how ? $\nearrow 0$ or $\searrow 0$ ?

3. Dec 8, 2009

Staff: Mentor

Split it up into two limits and work with them. If both limits exist, then their sum will be the limit you want.

Since both expressions involve exponents, you are most likely going to have to take the log, which will probably get you something you can use L'Hopital's Rule on.

4. Dec 9, 2009

abhineetK

I discussed this problem with someone..
He says,
lim (sinx)1/x=1
x->0
So, according to him answer must be 2.

I worked it out again and got the same answer, i.e., infinity...
How is it happening...??
Please, show the steps if you have got the answer 2, 1 or 0.
These were the options given to me...
However, if you feel that the correct answer is not among the options please tell with explanation and steps.

5. Dec 9, 2009

Staff: Mentor

I believe the limit in your first post is 1, but I can't justify it just yet. (sin x)^(1/x) -->0 as x -->0+, and (1/x)^(sin x) --> 1 as x --> 0+. I'll see if I can provide some justification for this result.

6. Dec 9, 2009

Staff: Mentor

As x approaches 0 from above. (The OP said that x > 0.)

7. Dec 9, 2009

Staff: Mentor

Let y1 = (1/x)sin x
Then ln y1 = ln[(1/x)sin x] = sinx*ln(1/x) = sinx*(-lnx) = -lnx/cscx
In the next step I use L'Hopital's Rule, since we have an indeterminate form of [infinity/infinity]. Also, in the following steps, lim means limit as x -->0+
lim ln[ ln y1] = lim [-lnx/cscx]
= lim [(-1/x)/(-csc x * cot x)] = lim[(1/x)/(csc x * cot x)]
= lim[(1/x)/(1/sinx * cos x/sinx)] = lim [sin^2(x)/(x cos x)]
= lim [(sin(x)/x * tan(x)] = lim[sin(x)/x]*lim tan(x) = 1*0 = 0.

Since lim ln y1 = 0, ln lim y1 = 0, hence lim y1 = 1.

The other limit is easier.
Let y2 = sinx1/x = (eln(sinx))1/x = e(1/x)ln(sin x)

Before taking the limit, let's look at what the factors in the exponent on e are doing as x --> 0+.

1/x --> + infinity.
sin(x) --> 0+, so ln(sin(x)) --> -infinity.
The product of 1/x and ln(sin(x)) --> -infinity.
Since the exponent on e is approaching -infinity, then e to that power --> 0.
Hence lim y2 = lim e(1/x)ln(sin x) = 0.

Therefore lim (y1 + y2) = 1 + 0 = 1.