Limit of monotic transformation

[tunaaa]

Hello, I was just wondering if is it true that the limit of a monotonic transformation of a function is the same as the monotonic transformation of its limit? That is, does

$$\lim_{n \rightarrow a} f(g(x)) = f(\lim_{n \rightarrow a} g(x))$$

for monotonic $$f$$, some $$a$$, and such that if the limit does not exist for one side of the expression, it doesn't exist for both?

Thanks.

 

[mathman]

Your original equation does not describe the dependence on n of anything.

 

[tunaaa]

Sorry, replace n with x - was very tired!

 

[tunaaa]

Correction: $$\lim_{x \rightarrow a} f(g(x)) = f(\lim_{x \rightarrow a} g(x))$$

 

[mathman]

I believe you need f to be continuous at g(a). The g(x) may be a red herring. Just look at lim(x->a)f(x)=f(a).