Evaluate lim_{x→0} arctan x / x.

Evaluate lim_{x→0} arctan x / x.

• A. 0
• B. 1
• C. ∞
• D. -1

Answer: (B)

The idea is that arctan x behaves like x very close to zero. Its slope at zero is arctan′(0) = 1, so for small x, arctan x ≈ x. This makes the ratio arctan x / x approach 1 as x → 0. You can see this more formally in a couple of standard ways. Using L’Hôpital’s rule on the 0/0 form, the derivative of the top is 1/(1+x^2) and the derivative of the bottom is 1, giving 1/(1+0) = 1. Alternatively, the Taylor series arctan x = x − x^3/3 + … shows arctan x / x = 1 − x^2/3 + …, which tends to 1 as x → 0. Thus the limit is 1.

The idea is that arctan x behaves like x very close to zero. Its slope at zero is arctan′(0) = 1, so for small x, arctan x ≈ x. This makes the ratio arctan x / x approach 1 as x → 0.

You can see this more formally in a couple of standard ways. Using L’Hôpital’s rule on the 0/0 form, the derivative of the top is 1/(1+x^2) and the derivative of the bottom is 1, giving 1/(1+0) = 1. Alternatively, the Taylor series arctan x = x − x^3/3 + … shows arctan x / x = 1 − x^2/3 + …, which tends to 1 as x → 0.

Thus the limit is 1.