Which expression denotes a limit taken with x approaching 0 from the positive side?

Which expression denotes a limit taken with x approaching 0 from the positive side?

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

Answer: (D)

Approaching zero from the positive side means x stays positive and gets arbitrarily close to zero. The expression that explicitly uses x → 0+ is the one that denotes this one-sided limit. Here, sqrt(x)/x simplifies to 1/√x, which grows without bound as x → 0+, so the limit is +∞. This shows how the plus in the limit notation signals a direction of approach, which is exactly what the question is testing. The other expressions describe two-sided limits and, in fact, approach finite values as x → 0 from either side (tan x / x → 1, (1 − cos x)/x^2 → 1/2, arctan x / x → 1).

Approaching zero from the positive side means x stays positive and gets arbitrarily close to zero. The expression that explicitly uses x → 0+ is the one that denotes this one-sided limit. Here, sqrt(x)/x simplifies to 1/√x, which grows without bound as x → 0+, so the limit is +∞. This shows how the plus in the limit notation signals a direction of approach, which is exactly what the question is testing. The other expressions describe two-sided limits and, in fact, approach finite values as x → 0 from either side (tan x / x → 1, (1 − cos x)/x^2 → 1/2, arctan x / x → 1).