Question

question 14
let $f(x)=\

$$\begin{cases} -\\frac{45}{x+5} & \\text{if } x < 4 \\\\ -11 & \\text{if } x = 4 \\\\ \\frac{15}{x-7} & \\text{if } x > 4 \\end{cases}$$

$
compute the quantities below. write \dne\ if the limit does not exist or the value is undefined.
$\lim_{x \to 4^-} f(x) = $
$\lim_{x \to 4^+} f(x) = $
$f(4) = $
since the above three quantities are $\boldsymbol{\text{select an answer}}$, we know that $f$ is $\boldsymbol{\text{select an answer}}$ at $x=4$.
list all numbers at which $f$ is discontinuous. be sure to check the functions defined to the left and right of 4 for discontinuities.

Explanation:

Step1: Find left-hand limit

Substitute $x=4$ into the left piece:
$\lim_{x \to 4^-} f(x) = -\frac{45}{4+5}$
$\lim_{x \to 4^-} f(x) = -\frac{45}{9} = -5$

Step2: Find right-hand limit

Substitute $x=4$ into the right piece:
$\lim_{x \to 4^+} f(x) = \frac{15}{4-7}$
$\lim_{x \to 4^+} f(x) = \frac{15}{-3} = -5$

Step3: Identify $f(4)$

From the piecewise function, $f(4) = -11$

Step4: Check continuity at $x=4$

A function is continuous at $x=a$ if $\lim_{x \to a^-}f(x) = \lim_{x \to a^+}f(x) = f(a)$. Here, $\lim_{x \to 4}f(x) = -5
eq f(4)=-11$, so $f$ is discontinuous at $x=4$.
Also, check for other discontinuities:

• For $x<4$: $-\frac{45}{x+5}$ is undefined at $x=-5$ (denominator 0)
• For $x>4$: $\frac{15}{x-7}$ is undefined at $x=7$ (denominator 0)

Answer:

$\lim_{x \to 4^-} f(x) = -5$
$\lim_{x \to 4^+} f(x) = -5$
$f(4) = -11$
Since the above three quantities are not equal, we know that $f$ is discontinuous at $x=4$.
List ALL numbers at which $f$ is discontinuous: $-5, 4, 7$