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

[tex]\lim_{x \to \infty} \sin x \cdot \sin \frac {1}{x}[/tex]

We say that

[tex] t=\frac{1}{x} [/tex]

and then:

[tex]\lim_{x \to \infty} \sin x \cdot \sin \frac {1}{x} = \lim_{t \to 0^+} \sin \frac{1}{t} \cdot \sin t[/tex]

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!

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# Variable change in limits

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