Let h(x)= x^ 3 +ax^ 2 +bx+c x^ 2 -9

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let

h(x)=\frac{x^{3}+ax^{2}+bx+c}{x^{2}-9},

where

a

b

, and

c

are real constants. Suppose that

\displaystyle\lim_{x\to 3}h(x)=4

exists and is finite. (1) Derive an equation relating

a

b

, and

c

that must be satisfied so that the given limit is finite. (2) Assume

c=27

. Find specific values of

a

and

b

that satisfy the relation from part (a) and that make the limit equal to

4

. (3) Verify directly that your choices of

a

and

b

in part (b) produce

\displaystyle \lim_{x\to 3}h(x)=4

. Show every algebraic step in your verification.