Limit of compositions at infinity.

  • Thread starter Yuqing
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We often have

[tex]lim_{x\rightarrow a}f(g(x)) = f(lim_{x\rightarrow a} g(x))[/tex]

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

But then my question arises where [tex]g(x)\rightarrow\infty[/tex]. 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?
 
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Delta2
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That equality holds if f(x) is continuous at [tex]lim_{x\rightarrow a}g(x)[/tex] (which is g(a) if g is continuous at a).

If [tex]lim_{x\rightarrow a}g(x)=\infty[/tex] then the equality isnt well defined, in this case it is correct to write that

[tex]lim_{x\rightarrow a}f(g(x))=lim_{x\rightarrow\infty}f(x)[/tex].

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.
 
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