Evaluate the limit as x approaches 0 of arctan x divided by x.

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Multiple Choice

Evaluate the limit as x approaches 0 of arctan x divided by x.

Explanation:
The limit is governed by the derivative at zero. Arctan is differentiable at 0, and its derivative there is arctan'(0) = 1/(1+0^2) = 1. By the definition of the derivative, the limit of [arctan x − arctan 0]/(x − 0) as x → 0 equals 1. Since arctan 0 = 0, this is exactly the limit of arctan x / x. Therefore, the limit is 1.

The limit is governed by the derivative at zero. Arctan is differentiable at 0, and its derivative there is arctan'(0) = 1/(1+0^2) = 1. By the definition of the derivative, the limit of [arctan x − arctan 0]/(x − 0) as x → 0 equals 1. Since arctan 0 = 0, this is exactly the limit of arctan x / x. Therefore, the limit is 1.

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