Evaluate the limit as x approaches 0 of arcsin(x) divided by x.

Evaluate the limit as x approaches 0 of arcsin(x) divided by x.

• A. 0
• B. 1
• C. 2
• D. -1

Answer: (B)

As x gets close to zero, arcsin x behaves like x because the inverse of sine mirrors the small-angle behavior of sine itself. If you set y = arcsin x, then x = sin y and y → 0 as x → 0. The limit becomes y / sin y as y → 0. Since sin y ~ y for small y, that ratio tends to 1. Another way to see this is the derivative of arcsin at zero: d/dx arcsin x = 1/√(1 - x^2), which equals 1 at x = 0, so the limit of arcsin x / x as x → 0 is 1. Therefore the limit is 1.

As x gets close to zero, arcsin x behaves like x because the inverse of sine mirrors the small-angle behavior of sine itself. If you set y = arcsin x, then x = sin y and y → 0 as x → 0. The limit becomes y / sin y as y → 0. Since sin y ~ y for small y, that ratio tends to 1. Another way to see this is the derivative of arcsin at zero: d/dx arcsin x = 1/√(1 - x^2), which equals 1 at x = 0, so the limit of arcsin x / x as x → 0 is 1. Therefore the limit is 1.