New variables and limits like xsin(1/x)

[shooba]

I'm supposed to find the limit, as x approaches infinity, of xsin(1/x).
I know that

xsin(1/x)=[sin(1/x)]/(1/x),

And that if I define t=(1/x), then,

As x approaches infinity, t approaches 0 from the right.

If I say that the original limit equals the limit, as t approaches 0 from the right, of (sint)/t, then, the limit equals 1, the correct answer.

However, I cannot justify the last step, this substitution and "transfer" of the limiting process to the new variable. It makes sense from an intuitive point of view, but I can't figure out how to prove it from basic concepts.

 

[HallsofIvy]

I'm not sure what it is you want to "justify".

You are asked to find
$$\lim_{x\to\infty}x sin(1/x)$$.

If you let t=1/x, then x= 1/t and sin(1/x)= sin(t) so the function becomes
$$\frac{sin(t)}{t}$$
and, of course, as x goes to infinity, t goes to 0.

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

There is nothing more to be said.

 

[shooba]

In general terms, what I'm trying to "justify" is that if the limit, as x approaches a, of g(x) is b, and we define t=g(x), then the limit, as x approaches a, of f(g(x)) is equal to the the limit, as t approaches b, of f(t).

In my case t=g(x)=1/x, f(t)=(sint)/t, a "=" infinity, and b=0.