# Problem of the week, 150

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.

Let a, b, and c be real numbers and let f and g be real valued functions of a real variable such that $\lim_{x\to a} g(x) = b$ and $\lim_{x\to b} f(x) = c$.
a. Give an example in which $\lim_{x\to a} f(g(x)) \not= c$.
b. Give an additional condition on f alone and show that it
guarantees $\lim_{x\to a} f(g(x)) = c$.
c. Give an additional condition on g alone and show that it
guarantees $\lim_{x\to a} f(g(x)) = c$.