Question
Determine \( \lim _{x \rightarrow-1^{-}} f(x) \) where
\[ f(x)=\left\{\begin{array}{ll}-5 & \text { if } x \leq-1 \\ 3 x-7 & \text { if }-11\end{array}\right. \]
If the answer in infinity, you can type in INF. If the answer is negative infinity, you
can type -INF. If the answer is does not exist, you can type in DNE. Otherwise type
in the appropriate number.
Ask by Olson Hobbs. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The limit is -5.
Solution
To determine the limit \( \lim _{x \rightarrow-1^{-}} f(x) \) for the given piecewise function:
\[
f(x)=\left\{\begin{array}{ll}
-5 & \text{if } x \leq -1 \\
3x - 7 & \text{if } -1 < x \leq 1 \\
7x^{2} & \text{if } x > 1
\end{array}\right.
\]
we need to consider the behavior of \( f(x) \) as \( x \) approaches \(-1\) from the left (i.e., \( x \) approaches \(-1\) with values less than \(-1\)).
For \( x \leq -1 \), the function \( f(x) \) is defined as a constant value:
\[
f(x) = -5
\]
Since \( f(x) \) is constant for all \( x \leq -1 \), as \( x \) approaches \(-1\) from the left, the value of \( f(x) \) remains \(-5\).
Therefore, the limit is:
\[
\lim _{x \rightarrow-1^{-}} f(x) = -5
\]
**Answer:**
-5
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find \( \lim _{x \rightarrow-1^{-}} f(x) \), we need to evaluate the function \( f(x) \) as \( x \) approaches \(-1\) from the left. According to the definition of the function: - For \( x \leq -1 \), \( f(x) = -5 \). - For \( -1 < x \leq 1 \), \( f(x) = 3x - 7 \). - For \( x > 1 \), \( f(x) = 7x^{2} \). Since we are looking at \( x \) approaching \(-1\) from the left, we use the first piece of the function, which applies when \( x \leq -1\): \[ \lim _{x \rightarrow-1^{-}} f(x) = -5. \] Thus, the answer is \(-5\).
