Consider the rational function defined for all real .

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

Problem

Consider the rational function

f(x)=\dfrac{3x+5}{x-2}

defined for all real

x\neq 2

. (a) Describe the sign of the numerator

3x+5

x

approaches

2

from the left and from the right. (b) Describe the sign of the denominator

x-2

x

approaches

2

from the left and from the right. (c) Using your answers to parts (a) and (b), determine the sign of

f(x)

x

approaches

2^{-}

and as

x

approaches

2^{+}

. Then state the value of each one-sided limit

\displaystyle \lim_{x\to 2^{-}} f(x)

and

\displaystyle \lim_{x\to 2^{+}} f(x)

. (d) Identify the vertical asymptote of

f

and give a concise verbal description of how the graph of

f

behaves near that asymptote on each side.