A rational function is defined by g(x)= (3k+2)x^ 6 -7x^ 2 +9 6x^ 5 +kx^ 4 -8

MATHEMATICS · Subjective · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

A rational function is defined by

g(x)=\frac{(3k+2)x^{6}-7x^{2}+9}{6x^{5}+kx^{4}-8},

where

k

is an integer. (i) Determine the limit

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

whenever it exists as a finite real number, expressing your answer in terms of

k

. (ii) Find all integer values of

k

for which the above limit is finite. (iii) For those

k

identified in part (ii), state whether the limit is

0

or a non-zero constant, giving justification in each case.