Full solution

Q. Determine whether the function

f(x)

is continuous at

x=-2

\newline

f(x)=\left\{\begin{array}{ll} 12-3 x^{2}, & x \leq-2 \\ -5-x, & x>-2 \end{array}\right.

\newline

f(x)

is continuous at

x=-2

\newline

f(x)

is discontinuous at

x=-2

Check Conditions: To determine if the function $f(x)$ is continuous at $x=-2$ , we need to check if the following three conditions are met: $\newline$ $1$ . $f(-2)$ is defined. $\newline$ $2$ . The limit of $f(x)$ as $x$ approaches $-2$ from the left side ( $\lim_{x\to-2^-} f(x)$ ) exists. $\newline$ $3$ . The limit of $f(x)$ as $x$ approaches $-2$ from the right side ( $x=-2$ $0$ ) exists and is equal to the limit from the left side and to $f(-2)$ . $\newline$ First, we will find $f(-2)$ using the piece of the function defined for $x=-2$ $3$ .
Find $f(-2)$ : Substitute $x = -2$ into the first piece of the function, which is $12 - 3x^2$ , to find $f(-2)$ .
$f(-2) = 12 - 3(-2)^2$
$f(-2) = 12 - 3(4)$
$f(-2) = 12 - 12$
$f(-2) = 0$
So, $f(-2)$ is defined and equals $0$ .
Left Side Limit: Next, we need to find the limit of $f(x)$ as $x$ approaches $-2$ from the left side. Since the function for $x \leq -2$ is $12 - 3x^2$ , we use this expression to find the limit. $\newline$ $\lim_{x\to-2^-} f(x) = \lim_{x\to-2^-} (12 - 3x^2)$ $\newline$ $\lim_{x\to-2^-} f(x) = 12 - 3(-2)^2$ $\newline$ $\lim_{x\to-2^-} f(x) = 12 - 12$ $\newline$ $\lim_{x\to-2^-} f(x) = 0$ $\newline$ The limit from the left side exists and equals $0$ .
Right Side Limit: Now, we need to find the limit of $f(x)$ as $x$ approaches $-2$ from the right side. Since the function for $x > -2$ is $-5 - x$ , we use this expression to find the limit. $\newline$ $\lim_{x\to-2^+} f(x) = \lim_{x\to-2^+} (-5 - x)$ $\newline$ $\lim_{x\to-2^+} f(x) = -5 - (-2)$ $\newline$ $\lim_{x\to-2^+} f(x) = -5 + 2$ $\newline$ $\lim_{x\to-2^+} f(x) = -3$ $\newline$ The limit from the right side exists and equals $-3$ .
Function Continuity: Since the limit from the left side ( $0$ ) is not equal to the limit from the right side ( $-3$ ), the function $f(x)$ is not continuous at $x=-2$ . For a function to be continuous at a point, the limit from the left must equal the limit from the right and also equal the function value at that point. In this case, the limits do not match.