Consider the piecewise–defined function f(x)= cases x^ 2 -9 x-3 , & x3, L, & x=3

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

Problem

Consider the piecewise–defined function

f(x)=\begin{cases}\dfrac{x^{2}-9}{x-3}, & x<3,\\ \dfrac{x^{2}-9a}{x-3}, & x>3,\\ L, & x=3,\end{cases}

where

a

and

L

are real constants. Tables of experimental function values on either side of

x=3

are provided. Left–hand data (

x<3

)

$x$	$2.80$	$2.90$	$2.99$	$2.999$
$f(x)$	$5.80$	$5.90$	$5.99$	$5.999$

Right–hand data (

x>3

)

$x$	$3.10$	$3.01$	$3.001$
$f(x)$	$6.10$	$6.01$	$6.001$

(a) Use the two tables to estimate the one–sided limits

\displaystyle\lim_{x\to3^-}f(x)

and

\displaystyle\lim_{x\to3^+}f(x)

. Record each limit correct to the nearest

0.001

. (b) Determine the value of

a

for which

\displaystyle\lim_{x\to3}f(x)

exists. Justify your answer using appropriate reasoning. (c) For the value of

a

found in part (b), evaluate

\displaystyle\lim_{x\to3}f(x)

. (d) Find

L

so that

f

is continuous at

x=3

. (e) Provide a brief justification that your choices of

a

and

L

guarantee continuity of

f

x=3