Suppose g(x) → a as x → a, and f(t) = 0 for t ≠ a, f(a) = 1. What is lim_{x→a} f(g(x))?

Suppose g(x) → a as x → a, and f(t) = 0 for t ≠ a, f(a) = 1. What is lim_{x→a} f(g(x))?

• A. 0
• B. 1
• C. a
• D. undefined

Answer: (A)

As x approaches a, the inner input g(x) gets arbitrarily close to a. The function f is zero for every input except at a, so whenever g(x) is not exactly a, f(g(x)) equals 0. The limit cares about values as x nears a, not at the point x = a itself, so these near-a values drive the limit. The fact that f(a) = 1 does not affect the limit, since it depends on the values of f at inputs approaching a, not on the value exactly at a. Therefore the limit of f(g(x)) as x→a is 0.

As x approaches a, the inner input g(x) gets arbitrarily close to a. The function f is zero for every input except at a, so whenever g(x) is not exactly a, f(g(x)) equals 0. The limit cares about values as x nears a, not at the point x = a itself, so these near-a values drive the limit. The fact that f(a) = 1 does not affect the limit, since it depends on the values of f at inputs approaching a, not on the value exactly at a. Therefore the limit of f(g(x)) as x→a is 0.