Given the following piecewise function, evaluate limx→1−f(x).

f(x)=⎧⎨⎩−2x2−2xx<−23x−2−2<x<1x2+3x>1

To evaluate the left-hand limit \(\lim_{x \to 1^-} f(x)\) for the given piecewise function, we need to find which piece of the function corresponds to the case when \(x\) approaches 1 from the left side.

The given piecewise function is:

\[
f(x) = 
\begin{cases} 
-2x^2 - 2x & \text{for } x < -2 \\
3x - 2 & \text{for } -2 < x < 1 \\
x^2 + 3 & \text{for } x > 1 
\end{cases}
\]

Since we're interested in \(\lim_{x \to 1^-} f(x)\), the relevant expression for \(f(x)\) is the piece defined for \( -2 < x < 1 \), which is:

\[
f(x) = 3x - 2.
\]

Now, take the limit as \(x\) approaches 1 from the left:

\[
\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (3x - 2).
\]

Substitute \(x = 1\):

\[
\lim_{x \to 1^-} f(x) = 3(1) - 2 = 3 - 2 = 1.
\]

Thus, the left-hand limit is:

\[
\lim_{x \to 1^-} f(x) = 1.
\]

Would you like further details on the limit calculation or how piecewise functions work?

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Here are 5 related questions:

1. How do you compute right-hand limits for piecewise functions?
2. What is the definition of a two-sided limit?
3. How do you determine if a function is continuous at a point?
4. What is the significance of approaching a limit from the left (\(x \to a^-\)) or right (\(x \to a^+\))?
5. How can you evaluate limits that involve more complex functions, such as logarithmic or trigonometric functions?

**Tip:** When evaluating limits, always check which part of a piecewise function is valid for the region you are approaching from!

Given the following piecewise function, evaluate lim x→1− f(x). f(x) = {−2x^2−2x for x < −2, 3x−2 for −2 < x < 1, x^2 + 3 for x > 1}

Learn how to evaluate the left-hand limit of a piecewise function as x approaches 1. This problem involves key calculus concepts such as limits, piecewise functions, and substitution, and provides a step-by-step solution. Suitable for pre-calculus or early calculus students in Grades 11-12.

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