Mathematics

Problem

A function

g

is defined by

g(x)=\begin{cases} \dfrac{x^2-ax+b}{x-3}, & x<3,\\ \sqrt{cx+d}, & x\ge 3, \end{cases}

where

a

b

c

, and

d

are real parameters satisfying

c>0

. a. Part 1: (a) Derive two independent equations relating

a

b

c

, and

d

that must be satisfied so that

\displaystyle \lim_{x\to 3} g(x)

exists and equals

5

. b. Part 2: (b) Given that

d=1

and

c=4

, solve for

a

and

b

using the equations obtained in part (a). c. Part 3: (c) Verify that your values of

a

and

b

indeed produce the required limit by computing the left–hand and right–hand limits explicitly. d. Part 4: (d) Discuss how the degree of the numerator in the first branch influences the strategy you used to create the equations in part (a).

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