# The limit of a composite function

I am presented with the following task:

"You are given that $k(h) \neq 0$ and $h \neq 0$. If $\lim_{k \to 0} F(k) = L$ and $\lim_{h \to 0} k(h) = 0$, show that $\lim_{h \to 0}F(k(h)) = L$.

This is the first part of a task that the professor marked with "Hard and theoretical", so I assume that it gets more complicated as we go, but I am wondering if my logic here is correct:

Given that $\lim_{h \to 0} k(h) = 0$, using $k(h)$ as the "inner part" (if there's a better word for this, let me know) of a composite function makes no difference for the result, thus $\lim_{h \to 0}F(k(h)) = L$. This is under the prerequisite that $k(h) \neq 0$, which we are already given.

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Are the functions $F$ and $k$ continuous? –  Jack Dawkins Sep 24 '13 at 7:15
That information is not provided. –  Andrew Thompson Sep 24 '13 at 7:18
Your logic is incorrect. Try to write justification for every step you say. For example, what do you mean by saying that the "inner part" does not matter? –  Vishal Sep 24 '13 at 7:27
Anyway, the result is not true: let $k(x) = x$ and $f(x) = \sin(1/x)$ if $x \neq 0$ and $f(0) = 0$. Continuity of $f$ is essential. –  Vishal Sep 24 '13 at 7:29