The graph of a function is described on the interval as follows

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

Problem

The graph of a function

g

is described on the interval

4<x<7

as follows. • For

4<x<5

, the graph is a branch of

y=\dfrac{2}{x-5}

, decreasing without bound as

x\to 5^-

. The points

(4.5,4)

and

(4.9,20)

lie on the curve. • At

x=5

g

is undefined and the line

x=5

is a vertical asymptote. • For

5<x<6

, the graph oscillates between

y=1

and

y=-1

with ever-decreasing amplitude so that

|g(x)|<\dfrac{1}{x-5}

for these

x

. • At

x=6

, the graph includes a solid point

(6,3)

and no other point directly above

x=6

. • For

6<x<7

, the graph is the line passing through

(6,3)

with slope

-2

. Assume no other features occur on

4<x<7

. (a) Determine

\displaystyle \lim_{x\to 5^-} g(x)

. Provide a clear justification based on the stated behavior of the graph. (b) Determine

\displaystyle \lim_{x\to 5^+} g(x)

. Explain why the oscillatory description implies your answer. (c) Does

\displaystyle \lim_{x\to 5} g(x)

exist? Support your answer with appropriate reasoning. (d) Evaluate

\displaystyle \lim_{x\to 6^-} g(x)

and

\displaystyle \lim_{x\to 6^+} g(x)

, then decide whether

\displaystyle \lim_{x\to 6} g(x)

exists. Justify each statement.