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Define f(x) = lim_(n->inf) (2*x^(2n) + x + 2) / (x^(2n) - x² + 3), where n is a positive integer. Which of the following is correct?

1. f(x) is continuous at x = 1
2. f(x) is discontinuous at x = 1
3. lim_(x->2) f(x) exists and is finite
4. lim_(x->-2) f(x) exists and is finite

Correct answer: f(x) is discontinuous at x = 1

Solution

For |x|<1, f(x)=(x+2)/(3-x²); as x->1⁻, f->3/2. For |x|>1, f=2; as x->1⁺, f->2. Since left and right limits differ (3/2 vs 2), f is discontinuous at x=1. At x=2 and x=-2 (both |x|>1), f(x)=2, so the limits exist finitely.

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