If limit of f is L and limit of g is M, then limit of g composed f is M?

by SebastianLinde   Last Updated November 08, 2018 23:20 PM

Problem:

Find examples of functions $f$ and $g$ defined on $\mathbb{R}$ with $\lim\limits_{x\to a}f(x) = L$, $\lim\limits_{y\to L}g(y) = M$, and $\lim\limits_{x\to a} g(f(x))\neq M$.

I have tried various combinations like $f(x) = x$ and $g(y) = y^2$, $f(x) = b$ and $g(y) = y^2$, and so on. I have even tried with some trigonometric functions with no luck. I am wondering what characteristic am I trying to "break" so that the conditions do not hold. Also, since $f$ and $g$ have to be defined on $\mathbb{R}$, does that mean that something like $\frac{1}{x}=f(x)$ is not a valid example since it is not defined at $x=0$?

Thanks for your help.



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Updated June 25, 2018 17:20 PM