Which limit diverges to infinity?

Which limit diverges to infinity?

• A. lim x->0 (1 - cos x)/x^2
• B. lim x->0 (arctan x)/x
• C. lim x->∞ (2x^2 - x)/(x^2 + 4x + 1)
• D. lim x->0+ sqrt(x)/x

Answer: (D)

This question tests how to tell when a limit grows without bound by watching the variable approach the specified point from the indicated direction. For the limit as x approaches 0 from the positive side of sqrt(x) divided by x, rewriting helps: sqrt(x)/x = 1/√x. As x gets smaller and positive, 1/√x grows without bound, so the limit goes to infinity. To see why the other options don’t diverge, note that (1 − cos x)/x^2 approaches 1/2 as x → 0, not infinity; arctan x behaves like x near 0, so (arctan x)/x → 1; and for large x the expression (2x^2 − x)/(x^2 + 4x + 1) behaves like 2x^2/x^2 → 2.

This question tests how to tell when a limit grows without bound by watching the variable approach the specified point from the indicated direction. For the limit as x approaches 0 from the positive side of sqrt(x) divided by x, rewriting helps: sqrt(x)/x = 1/√x. As x gets smaller and positive, 1/√x grows without bound, so the limit goes to infinity.

To see why the other options don’t diverge, note that (1 − cos x)/x^2 approaches 1/2 as x → 0, not infinity; arctan x behaves like x near 0, so (arctan x)/x → 1; and for large x the expression (2x^2 − x)/(x^2 + 4x + 1) behaves like 2x^2/x^2 → 2.