Compute lim_{x->0} (arctan x)/x.

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Multiple Choice

Compute lim_{x->0} (arctan x)/x.

Explanation:
The limit tests how arctan x scales with x near zero, essentially reading off its slope at 0. The derivative of arctan x is 1/(1+x^2), so at x = 0 the slope is 1. Since arctan 0 = 0, the ratio arctan x / x tends to that slope as x → 0. Equivalently, arctan x has the expansion arctan x = x − x^3/3 + o(x^3), and dividing by x gives 1 − x^2/3 + o(x^2) → 1. The limit is 1.

The limit tests how arctan x scales with x near zero, essentially reading off its slope at 0. The derivative of arctan x is 1/(1+x^2), so at x = 0 the slope is 1. Since arctan 0 = 0, the ratio arctan x / x tends to that slope as x → 0. Equivalently, arctan x has the expansion arctan x = x − x^3/3 + o(x^3), and dividing by x gives 1 − x^2/3 + o(x^2) → 1. The limit is 1.

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