For the function f(x)= cases x x , & x 0, 1, & x=0

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

Problem

For the function f(x)={sin⁡xx,x≠0,1,x=0,f(x)=\begin{cases}\dfrac{\sin x}{x}, & x\neq 0,\\1, & x=0,\end{cases} consider values of xx satisfying 0<x<π20<x<\dfrac{\pi}{2}. (1) Show algebraically that cos⁡x<sin⁡xx<1\cos x<\dfrac{\sin x}{x}<1 for all 0<x<π20<x<\dfrac{\pi}{2}. Provide a clear justification for each inequality sign. (2) Using the inequalities from part (a), determine lim⁡x→0f(x)\displaystyle \lim_{x\to 0} f(x). Give a complete explanation of your reasoning without appealing to derivatives or series expansions. (3) Explain why the result from part (b) implies that ff is continuous at x=0x=0. In your explanation, refer explicitly to the definition of continuity in terms of limits.

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