Consider the piecewise function p(x)= cases x^2-9 x-3 , & x 3

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

Problem

Consider the piecewise function

p(x)=\begin{cases} \dfrac{x^2-9}{x-3}, & x\neq3,\\ k, & x=3, \end{cases}

where

k

is a real constant. (1) Part 1: (a) Determine the value of

k

for which

p

is continuous at

x=3

. Show all reasoning. (2) Part 2: (b) Using the value of

k

found in part (a), compute

\displaystyle \lim_{x\to3}\frac{p(x)-p(3)}{x-3},

and interpret the result. (3) Part 3: (c) Verify numerically that the limit in part (b) approximates the slope of the secant line joining

(3,\,p(3))

and

(3.01,\,p(3.01))

. Provide appropriate calculations to support your claim.