Consider the function

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Consider the function f(x)=sin⁡xxf(x)=\dfrac{\sin x}{x}, where x≠0x\neq0. (i) Find f′(x)f'(x) and simplify the result as far as possible. (ii) Evaluate f′ ⁣(π4)f'\!\left(\dfrac{\pi}{4}\right). Express your answer in exact form. (iii) Verify that lim⁡x→0f′(x)\displaystyle \lim_{x\to0}f'(x) exists and state its value.

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