Question

graph the given function and then find the specified limits. if necessary, state that the limit does not exist.
$g(x) = x^2 - 3$, $limlimits_{x \to -2} g(x)$ and $limlimits_{x \to 3} g(x)$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. $limlimits_{x \to -2} g(x) = \square$ (simplify your answer.)
\\(\bigcirc\\) b. the limit does not exist and is neither $\infty$ nor $-\infty$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. $limlimits_{x \to 3} g(x) = \square$ (simplify your answer.)
\\(\bigcirc\\) b. the limit does not exist and is neither $\infty$ nor $-\infty$
choose the correct graph below
\\(\bigcirc\\) a. graph of a parabola opening upwards \\(\bigcirc\\) b. graph of a parabola opening downwards \\(\bigcirc\\) c. graph of a piecewise function \\(\bigcirc\\) d. graph of a linear function

Explanation:

Step1: Identify function type

$g(x)=x^2 - 3$ is a quadratic function (parabola opening upward, vertex at $(0,-3)$).

Step2: Match with graph

Graph A is an upward-opening parabola with vertex near $(0,-3)$, matching $g(x)=x^2 -3$.

Step3: Calculate $\lim_{x \to -2} g(x)$

Substitute $x=-2$ into $g(x)$: $(-2)^2 -3 = 4 -3 =1$.

Step4: Calculate $\lim_{x \to 3} g(x)$

Substitute $x=3$ into $g(x)$: $3^2 -3 =9 -3 =6$.

Answer:

Correct graph: A
$\lim_{x \to -2} g(x)$: A. $\lim_{x \to -2} g(x)=1$
$\lim_{x \to 3} g(x)$: A. $\lim_{x \to 3} g(x)=6$