Let h(x)= px^ 2 +qx+1 4x^ 2 -1

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

Problem

Let

h(x)=\frac{px^{2}+qx+1}{4x^{2}-1},

where

p

and

q

are real constants. (a) Find

\displaystyle\lim_{x\to\infty}h(x)

in terms of

p

. Show your steps clearly. (b) Determine all real values of

p

for which

\displaystyle\lim_{x\to\infty}h(x)=\tfrac{3}{2}

. Justify your conclusion algebraically. (c) For each value of

p

obtained in part (b), decide whether

\displaystyle\lim_{x\to-\infty}h(x)

equals

\tfrac{3}{2}

as well. Provide a thorough explanation.