Problem of the week, 150

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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.

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