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Let $f$ and $g$ be functions with domain $\mathbb{R}$. Suppose $\lim_{x \rightarrow a} f(x) = b$ and $\lim_{x \rightarrow b} g(x) = c$. Prove or disprove that $\lim_{x \rightarrow a} (g \circ f)(x) = c$. (If true, explain why, with a rigorous proof if possible; if false, give an example.)

 Jul 18, 2022
 #1
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A counter-example is given by f(x) = x*sin(1/x) and g(x) = x^2.

 Jul 18, 2022
 #2
avatar+118141 
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counter example

 

f(x)=x-2

g(x)=x^2-4

 

\(\displaystyle \lim_{x\rightarrow2}\;f(x)=2-2=0\\ \displaystyle \lim_{x\rightarrow2}\;g(x)=2^2-4=0\\ \displaystyle \lim_{x\rightarrow2}\;(f \circ g)(x)= \lim_{x\rightarrow2}\;( x^2-4-2 )= -2\\ \displaystyle \lim_{x\rightarrow2}\;(g \circ f)(x)= \lim_{x\rightarrow2}\;( (x-2)^2-4 )= -4\\\)

 Jul 19, 2022

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