Mathematics

Problem

A piecewise–defined function

h

is given by

h(x)=\begin{cases} \dfrac{a x^{2}+3}{2x^{2}-1}, & x\ge 0,\\ \dfrac{b x^{2}-5}{x^{2}+2}, & x<0, \end{cases}

where

a

and

b

are real constants. a. Part 1: (a) Find the values of

a

and

b

so that the horizontal asymptote of

h

x\to\infty

y=\dfrac{3}{4}

and the horizontal asymptote as

x\to-\infty

y=-2

. Justify your work. b. Part 2: (b) Using the values of

a

and

b

found in part (a), evaluate

\displaystyle\lim_{x\to0^-}h(x)

and

\displaystyle\lim_{x\to0^+}h(x)

. Does

h

remain continuous at

x=0

under your choices of

a

and

b

? Explain. c. Part 3: (c) Determine the value of

h(0)

(if any) that would make

h

continuous at

x=0

. Provide justification using the appropriate limit laws.