A rational function is defined by f(x)= 3x+5 x-2

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

A rational function is defined by

f(x)=\frac{3x+5}{x-2}, \quad x\neq 2.

Answer the following questions about the behavior of

f

near

x=2

. (a) Describe in words what it means to say that

\displaystyle \lim_{x\to 2}f(x)=\infty

. Your answer should reference the values of

x

and the corresponding values of

f(x)

. (b) Determine

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

and

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

. Show all steps leading to your conclusion. (c) State the equation of any vertical asymptote of

f

and provide a brief justification based on your work in part (b).