A function g is defined as follows: g(x) = max{x, 1/x} / min{x, 1/x} when x != 0, and g(0) = 1. Which of the following statements is correct?

lim(x->0+) g(x) = 0
lim(x->0-) g(x) = 1
g(x) is continuous for all x except at x = 0
g(x) is differentiable for all x except at x = 0

Correct answer: g(x) is continuous for all x except at x = 0

Solution

For 0 < x < 1, g(x) = 1/x², which tends to +infinity as x->0+, so the right-hand limit is not 0 or 1. Analysis shows g is continuous everywhere except x = 0, but fails to be differentiable at x = 1 (and x = -1) where the formula switches, so option D is incorrect.

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