Limit of compositions at infinity.

[Yuqing]

We often have

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

if f(x) is continuous at g(a).

But then my question arises where g(x)\rightarrow\infty. I am not sure if there is any meaning to continuity at infinity as it seems that continuity is the property of a particular point. If the function is proven to be continuous for all x or at least for large x then will this equality hold?

 

[Delta2]

That equality holds if f(x) is continuous at lim_{x\rightarrow a}g(x) (which is g(a) if g is continuous at a).

If lim_{x\rightarrow a}g(x)=\infty then the equality isn't well defined, in this case it is correct to write that

lim_{x\rightarrow a}f(g(x))=lim_{x\rightarrow\infty}f(x).

This last limit might exist or not and might be finite or infinite. For example for f(x)=x the limit gives infinite, for f(x)=1/x (f here is continuous in R-{0}) it is 0 and for f(x)=sinx it does not exist.

P.S Continuity is not defined at infinity.