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$g(x)={[(1)/(x)," for "x < -1],[1+2x," for "-1 <= x <= 0]:} Find lim_(x rarr-1)g(x). Choose 1 answer: (A) -1 (B) 1 (C) 3 (D) The limit doesn't exist.$

$g(x)=\left\{\begin{array}{ll}\frac{1}{x} & \text { for } x<-1 \\ 1+2 x & \text { for }-1 \leq x \leq 0\end{array}\right.$ $\newline$ Find $\lim _{x \rightarrow-1} g(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-1$ $\newline$ (B) $1$ $\newline$ (C) $3$ $\newline$ (D) The limit doesn't exist.

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Solve quadratic inequalities

Full solution

Q.

g(x)=\left\{\begin{array}{ll}\frac{1}{x} & \text { for } x<-1 \\ 1+2 x & \text { for }-1 \leq x \leq 0\end{array}\right.

\newline

Find

\lim _{x \rightarrow-1} g(x)

.

\newline

Choose

1

answer:

\newline

(A)

-1

\newline

(B)

1

\newline

(C)

3

\newline

(D) The limit doesn't exist.

Identify appropriate piece: Identify the appropriate piece of the piecewise function to use for the limit as $x$ approaches $-1$ . Since we are looking for the limit as $x$ approaches $-1$ , we need to determine which part of the piecewise function applies. The function $g(x)$ is defined as $\frac{1}{x}$ for x < -1 and as $1+2x$ for $-1 \leq x \leq 0$ . To find the limit as $x$ approaches $-1$ , we need to consider the value from both sides of $-1$ .
Calculate limit from left side: Calculate the limit from the left side of $-1$ . $\newline$ For $x$ approaching $-1$ from the left (x < -1), we use the first part of the piecewise function $g(x) = \frac{1}{x}$ . The limit as $x$ approaches $-1$ from the left is: $\newline$ $\lim_{x \to -1^-} \frac{1}{x}$ $\newline$ Since we cannot divide by zero, we need to consider the behavior of $\frac{1}{x}$ as $x$ gets very close to $-1$ from the left. As $x$ approaches $-1$ from the left, $\frac{1}{x}$ approaches negative infinity. However, since we are looking for the limit at exactly $-1$ , we need to consider the value from the right side as well.
Calculate limit from right side: Calculate the limit from the right side of $-1$ . For $x$ approaching $-1$ from the right $(-1 \leq x \leq 0)$ , we use the second part of the piecewise function $g(x) = 1+2x$ . The limit as $x$ approaches $-1$ from the right is: $\lim_{x \to -1^+} 1+2x$ Substitute $x$ with $-1$ : $1+2(-1) = 1-2 = -1$
Determine if limit exists: Determine if the limit exists by comparing the limits from the left and right sides of $-1$ . The limit from the left side as $x$ approaches $-1$ is negative infinity, and the limit from the right side as $x$ approaches $-1$ is $-1$ . Since these two limits do not match, the overall limit of $g(x)$ as $x$ approaches $-1$ does not exist.

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