Let and be differentiable functions defined on an open interval containing

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

Problem

Let ff and gg be differentiable functions defined on an open interval containing x=2x=2, except possibly at x=2x=2 itself. Assume • lim⁡x→2f(x)=0\displaystyle \lim_{x\to 2}f(x)=0 and lim⁡x→2g(x)=0\displaystyle \lim_{x\to 2}g(x)=0. • f′(2)=5f'(2)=5 and g′(2)=−2g'(2)=-2. Answer the following questions about the quotient f(x)g(x)\dfrac{f(x)}{g(x)}. (a) Explain why the information given so far is insufficient to decide immediately whether lim⁡x→2f(x)g(x)\displaystyle \lim_{x\to 2}\dfrac{f(x)}{g(x)} exists. (b) Determine the value of lim⁡x→2f(x)g(x)\displaystyle \lim_{x\to 2}\dfrac{f(x)}{g(x)} or state that it does not exist, justifying your conclusion. (c) Based on your result in part (b), indicate whether x=2x=2 is a vertical asymptote for the graph of y=f(x)g(x)y=\dfrac{f(x)}{g(x)}. Give a brief justification.

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