The function is defined piecewise by g(x)= cases 1 x-4 , & x<4, [4pt] x, & x 4

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

Problem

The function

g

is defined piecewise by

g(x)=\begin{cases} \dfrac{1}{x-4}, & x<4,\\[4pt] \sin x, & x\ge 4. \end{cases}

Analyze the behavior of

g

x=4

. (a) Determine

\displaystyle\lim_{x\to 4^-}g(x)

and show all supporting work. (b) Determine

\displaystyle\lim_{x\to 4^+}g(x)

and show all supporting work. (c) Does

\displaystyle\lim_{x\to 4}g(x)

exist? Justify your answer by referencing the results of parts (a) and (b). (d) Explain whether

x=4

corresponds to a vertical asymptote of the graph of

g

. Provide reasoning that cites the formal definition of an infinite limit where appropriate. (e) Describe, in words, the overall graphical behavior of

g

in a neighborhood of

x=4

. Your description should reference direction of unbounded growth, continuity, and any intersection with the point

(4,\sin 4)

. No sketch is required, but your explanation must be sufficiently detailed that another student could draw an accurate local graph from it.

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