Let f and g be functions with domain R. Suppose limx→af(x)=b and limx→bg(x)=c. Prove or disprove that limx→a(g∘f)(x)=c. (If true, explain why, with a rigorous proof if possible; if false, give an example.)
A counter-example is given by f(x) = x*sin(1/x) and g(x) = x^2.
counter example
f(x)=x-2
g(x)=x^2-4
limx→2f(x)=2−2=0limx→2g(x)=22−4=0limx→2(f∘g)(x)=limx→2(x2−4−2)=−2limx→2(g∘f)(x)=limx→2((x−2)2−4)=−4
