Evaluate the following limits if it exist

[haha1234]

Homework Statement

evaluate the following limits if it exist
lim x sin1/x and limit x sin 1/x
x→0 x→∞

Homework Equations

The Attempt at a Solution

Someone have told me that I should let t=1/x and rewrite the limits.However, once I rewrite the limit, I still cannot evaluate the limits.
So,how can I find the limits?

 

[CAF123]

First rewrite the expression as $$\lim_{x \rightarrow 0} \frac{\sin\left(1/x\right)}{1/x}$$ and then the reason for introducing the variable $t$ becomes more obvious.

 

[HallsofIvy]

You should have learned the limit $$\lim_{\theta\to 0}\frac{sin(\theta)}{\theta}$$ in first semester Calculus. If you do not remember it, look in a Calculus text in the section on the derivative of sin(x).

 

[Mark44]

Homework Statement

evaluate the following limits if it exist
lim x sin1/x and limit x sin 1/x
x→0 x→∞

Homework Equations

The Attempt at a Solution

Someone have told me that I should let t=1/x and rewrite the limits.However, once I rewrite the limit, I still cannot evaluate the limits.
So,how can I find the limits?

The OP's problem statement wasn't very clear, but there are two problems here.

$$\lim_{x \to 0} x sin(1/x)$$
$$\lim_{x \to \infty} x sin(1/x)$$

The hint applies to the second problem.

 

[haha1234]

You should have learned the limit $$\lim_{\theta\to 0}\frac{sin(\theta)}{\theta}$$ in first semester Calculus. If you do not remember it, look in a Calculus text in the section on the derivative of sin(x).

I have used this method,but I still cannot find correct answer that I got the answer is 1.

 

[haha1234]

First rewrite the expression as $$\lim_{x \rightarrow 0} \frac{\sin\left(1/x\right)}{1/x}$$ and then the reason for introducing the variable $t$ becomes more obvious.

Why limx→0xsin(1/x)is the same as limx→0sin(1/x)1/x ?:shy:

 

[Office_Shredder]

haha, the only thing they did to rewrite the limit like that is
$$x = \frac{1}{1/x}$$

 

[Mark44]

Why limx→0xsin(1/x)is the same as limx→0sin(1/x)1/x ?:shy:

It isn't the same. What you wrote is sin(1/x) * 1/x. What it should be is sin(1/x)/(1/x).