Let for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let f(x)=1+sin⁡x−1xf(x)=\dfrac{\sqrt{1+\sin x}-1}{x} for all real x≠0x\neq 0. (a) Compute lim⁡x→0f(x)\displaystyle\lim_{x\to 0} f(x), showing each algebraic and trigonometric step. (b) Interpret the meaning of your result in terms of the instantaneous rate of change of the function 1+sin⁡x\sqrt{1+\sin x} at x=0x=0.

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