Variable Change in Limits: When is it Permissible and Accurate?

[LeifEricson]

Hello,

I see that a common method to calculating limits is a change of the variable. For example, to calculate:

$$\lim_{x \to \infty} \sin x \cdot \sin \frac {1}{x}$$

We say that
$$t=\frac{1}{x}$$

and then:

$$\lim_{x \to \infty} \sin x \cdot \sin \frac {1}{x} = \lim_{t \to 0^+} \sin \frac{1}{t} \cdot \sin t$$

My question is:
When can we do that? When is it allowed? What are the conditions that this will be accurate and true? What do we have make sure before we can use that?

This is something I've pondered about for very long and haven't found an answer. I would really appreciate an explanation.

Thanks!

 

[HallsofIvy]

You just have to make sure that you are making the change correctly! If x= 1/t, or more generally, x= f(t), you just have to be sure you replace every "x" with f(t). The one point on which you have to be careful is replacing "$x\rightarrow a$" with "$t\rightarrow b$" where $\lim_{t\rightarrow b} f(t)= a$- which means, of course, that limit must exist.

 

[maze]

You may want to consider the epsilon-delta definition of limit in a few cases to convince yourself of why it works.