Question

graph f, locate all points of discontinuity, and discuss the behavior of f at these points.
f(x)=\begin{cases}-1 + x&\text{if }x < - 1\\5 - x&\text{if }xgeq - 1end{cases}

b. there are no points of discontinuity.
discuss the behavior of f at its point(s) of discontinuity.
find f(x) at any points of discontinuity. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. at the point(s) of discontinuity, the value(s) of the function f(x) is/are 6.
(use a comma to separate answers as needed.)
b. there are no points of discontinuity.
find (lim_{x
ightarrow c}f(x)), where c is/are the point(s) at which f(x) is discontinuous. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. (lim_{x
ightarrow c}f(x)=)
(use a comma to separate answers as needed.)
b. the limit does not exist.

Explanation:

Step1: Check left - hand limit at \(x = - 1\)

For \(x\lt - 1\), \(f(x)=-1 + x\). Then \(\lim_{x
ightarrow - 1^{-}}f(x)=\lim_{x
ightarrow - 1^{-}}(-1 + x)=-1+( - 1)=-2\).

Step2: Check right - hand limit at \(x = - 1\)

For \(x\geq - 1\), \(f(x)=5 - x\). Then \(\lim_{x
ightarrow - 1^{+}}f(x)=\lim_{x
ightarrow - 1^{+}}(5 - x)=5-( - 1)=6\).

Step3: Check function value at \(x = - 1\)

For \(x=-1\), \(f(-1)=5-( - 1)=6\).
Since \(\lim_{x
ightarrow - 1^{-}}f(x)
eq\lim_{x
ightarrow - 1^{+}}f(x)\), the function \(f(x)\) is discontinuous at \(x=-1\).

Step4: Find the limit as \(x

ightarrow - 1\)
The left - hand limit \(\lim_{x
ightarrow - 1^{-}}f(x)=-2\) and the right - hand limit \(\lim_{x
ightarrow - 1^{+}}f(x)=6\). Since they are not equal, \(\lim_{x
ightarrow - 1}f(x)\) does not exist.

Answer:

At the point of discontinuity \(x=-1\), the value of the function \(f(x)\) is \(6\). The limit \(\lim_{x
ightarrow - 1}f(x)\) does not exist. So for the first part about the value of the function at the point of discontinuity, the answer is A. At the point(s) of discontinuity, the value(s) of the function \(f(x)\) is/are \(6\). For the part about \(\lim_{x
ightarrow c}f(x)\) where \(c\) is the point of discontinuity, the answer is B. The limit does not exist.