Question

let $g(x)=\

$$\begin{cases}-6x^2 + x & \\text{if } x < 3 \\\\ a & \\text{if } x = 3 \\\\ 2x + 7 & \\text{if } x > 3\\end{cases}$$

$
a. determine the value of $a$ for which $g$ is continuous from the left at $3$
b. determine the value of $a$ for which $g$ is continuous from the right at $3$
c. is there a value of $a$ for which $g$ is continuous at $3$

a. the value of $a$ for which $g$ is continuous from the left at $3$ is $-51$ (simplify your answer )
b. the value of $a$ for which $g$ is continuous from the right at $3$ is $13$ (simplify your answer )
c. select the correct choice below and, if necessary, fill in the answer box within your choice
\\(\boldsymbol{\text{o a.}}\\) when $a = \square$, $g$ is continuous at $3$ because $\lim\limits_{x\to 3^-} g(x) = \square$ and $\lim\limits_{x\to 3^+} g(x) = \square$
\\(\boldsymbol{\text{o b.}}\\) there is no value of $a$ for which $g$ is continuous at $3$ because $\lim\limits_{x\to 3^-} g(x) = \square$ and $\lim\limits_{x\to 3^+} g(x) = \square$

Explanation:

Part c

To determine if \( g(x) \) is continuous at \( x = 3 \), we need \( \lim_{x \to 3^{-}} g(x) = \lim_{x \to 3^{+}} g(x) = g(3) = a \).

Step 1: Find \( \lim_{x \to 3^{-}} g(x) \)

For \( x < 3 \), \( g(x) = -6x^2 + x \). Substitute \( x = 3 \):
\[
\lim_{x \to 3^{-}} g(x) = -6(3)^2 + 3 = -6(9) + 3 = -54 + 3 = -51
\]

Step 2: Find \( \lim_{x \to 3^{+}} g(x) \)

For \( x > 3 \), \( g(x) = 2x + 7 \). Substitute \( x = 3 \):
\[
\lim_{x \to 3^{+}} g(x) = 2(3) + 7 = 6 + 7 = 13
\]

Step 3: Check continuity

For \( g(x) \) to be continuous at \( x = 3 \), \( \lim_{x \to 3^{-}} g(x) \) must equal \( \lim_{x \to 3^{+}} g(x) \), but \( -51
eq 13 \). So, there's no value of \( a \) that makes \( g(x) \) continuous at \( 3 \) because the left - hand limit (\( -51 \)) and the right - hand limit (\( 13 \)) are not equal.

Answer:

B. There is no value of \( a \) for which \( g \) is continuous at \( 3 \) because \( \lim_{x
ightarrow3^{-}}g(x)=\boldsymbol{-51} \) and \( \lim_{x
ightarrow3^{+}}g(x)=\boldsymbol{13} \)