Question

1. the piece - wise function f is defined below. the graph of a piece - wise function y = h(x) and selected values of a continuous function g are given. let $f(x)=\begin{cases}3x + 4&x < - 2\\h(x)&-2leq x < 4\\g(x)&4leq xleq8\\frac{-3x+cx^{2}}{2(x - 10)(x + 4)}&x > 8end{cases}$

x	4	5	6	7	8
g(x)	3	1	-1	-3	-5

a. at which x - value(s) on the interval (-2, 4) does the graph of h have a removable discontinuity?
b. if $lim_{x
ightarrowinfty}f(x)=6$, find the value of c.
c. is f continuous at x = - 2? justify your answer.

Explanation:

Step1: Analyze removable discontinuity

Check for holes in $y = h(x)$ graph on $(-2,4)$.

Step2: Analyze limit for $c$

For $x\to\infty$, focus on $f(x)=\frac{-3x + cx^{2}}{2(x - 10)(x + 4)}$, simplify and solve for $c$ using $\lim_{x\to\infty}f(x)=6$.

Step3: Check continuity at $x=-2$

Find $\lim_{x\to - 2^{-}}f(x)$ and $\lim_{x\to - 2^{+}}f(x)$ and check if they are equal.

Answer:

a. Analyze graph of $h(x)$ for holes.
b. First simplify $\lim_{x\to\infty}\frac{-3x + cx^{2}}{2(x - 10)(x + 4)}=\lim_{x\to\infty}\frac{-3/x + c}{2(1 - 10/x)(1 + 4/x)}=\frac{c}{2}=6$, so $c = 12$.
c. $\lim_{x\to - 2^{-}}f(x)=3(-2)+4=-2$, find $\lim_{x\to - 2^{+}}f(x)$ from $h(x)$ graph and check equality.