Mathematics

Problem

Consider three independent functions

g

h

, and

k

defined on real numbers with the following properties. Function

g

: •

g(x)=\frac{x^2-9}{x-3}

for

x\neq 3

;

g

is undefined at

x=3

. Function

h

: •

h

is defined for all real

x

. •

\lim_{x\to 1^-}h(x)=4

and

\lim_{x\to 1^+}h(x)=-2

. •

h(1)=7

. Function

k

: •

k(x)=\frac{1}{(x+2)^2}

for

x\neq -2

;

k

is undefined at

x=-2

. (a) For each function, describe whether the limit exists at the indicated point (

x=3

for

g

x=1

for

h

, and

x=-2

for

k

). Provide numerical values when they exist. (b) Classify the discontinuity at the indicated point for each function as removable, jump, or infinite. Give a clear justification for each classification based on the definition of limit and continuity. (c) For any function(s) above that possess a removable discontinuity, explain precisely how the function could be redefined at the indicated point to make it continuous there. a. Part (a): Evaluate and discuss the existence of the appropriate limits. b. Part (b): Classify the type of discontinuity for each function with justification. c. Part (c): Describe how to redefine the relevant function(s) to achieve continuity.

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