# Limit x*sin(x) , x->inf

1. Jun 22, 2009

### karkas

1. The problem statement, all variables and given/known data
The complete exercise is:

If $\lim_{x->\inf } \frac{f(x)-5x^2sin(x)}{(\sqrt (x^2+2))-x} = 7$

show that $\lim_{x->\inf} \frac{f(x)}{x} = 5$

2. Relevant equations
How do I show that $\lim_{x->\inf} xsinx =1$, because I run into it!

3. The attempt at a solution

I set K(x) = the fraction of the first limit and I solved for f(x) (x=0 excluded).

Then I have the limit $\lim_{x->\inf} \frac{f(x)}{x} = \lim_{x->\inf} K(x)*0 + 5 xsinx$.

Yet finally I reach the limit I spoke about in 2.

Last edited: Jun 22, 2009
2. Jun 22, 2009

### zcd

You can't, it eventually oscillates between +/- infinity. What exactly is f(x) in this context?

3. Jun 22, 2009

### karkas

Random function. It doesn't specify... Any other solutions?

4. Jun 22, 2009

### zcd

Maybe you're omitting part of the question?
If $\lim_{x->\inf } \frac{f(x)-5x^2sin(x)}{(\sqrt (x^2+2))-x}$
doesn't say anything because you're only giving the condition. Does the limit = something? Is the question asking you to find f(x) such that $\lim_{x->\inf} \frac{f(x)}{x} = 5$?

5. Jun 22, 2009

### karkas

Yes indeed I'll fix it.

No, it just wants me to prove the second limit equals 5.