Consider the function f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell short dash 8 x plus 1 end cell cell i f space x less than short dash 2 end cell row cell 7 minus x minus x squared end cell cell i f space x greater or equal than short dash 2 end cell end table close.

Evaluate limit as x rightwards arrow short dash 2 to the power of short dash of f left parenthesis x right parenthesis and limit as x rightwards arrow short dash 2 to the power of plus of f left parenthesis x right parenthesis.

To evaluate the limits of the piecewise function \( f(x) \) as \( x \) approaches \(-2\) from the left and the right, we need to consider the definition of \( f(x) \):

\[ f(x) = \begin{cases} 
-8x + 1 & \text{if } x < -2 \\
7 - x - x^2 & \text{if } x \geq -2 
\end{cases} \]

**1. Evaluating the left-hand limit as \( x \) approaches \(-2\):**

For \( x \) approaching \(-2\) from the left (\( x \to -2^- \)), we use the first piece of the function:

\[ f(x) = -8x + 1 \]

So, we need to evaluate:

\[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} (-8x + 1) \]

Substitute \( x = -2 \):

\[ \lim_{x \to -2^-} (-8x + 1) = -8(-2) + 1 = 16 + 1 = 17 \]

Therefore,

\[ \lim_{x \to -2^-} f(x) = 17 \]

**2. Evaluating the right-hand limit as \( x \) approaches \(-2\):**

For \( x \) approaching \(-2\) from the right (\( x \to -2^+ \)), we use the second piece of the function:

\[ f(x) = 7 - x - x^2 \]

So, we need to evaluate:

\[ \lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} (7 - x - x^2) \]

Substitute \( x = -2 \):

\[ \lim_{x \to -2^+} (7 - x - x^2) = 7 - (-2) - (-2)^2 = 7 + 2 - 4 = 5 \]

Therefore,

\[ \lim_{x \to -2^+} f(x) = 5 \]

In summary:

\[ \lim_{x \to -2^-} f(x) = 17 \]
\[ \lim_{x \to -2^+} f(x) = 5 \]

If you have any questions or need further details, please let me know. Here are some additional questions you might find interesting:

1. How do you determine the continuity of a piecewise function?
2. What are the differences between left-hand and right-hand limits?
3. How do you evaluate limits at infinity?
4. What does it mean for a function to be continuous at a point?
5. How do you use the epsilon-delta definition of a limit?

**Tip:** When evaluating piecewise functions at a specific point, always consider the corresponding piece of the function for the appropriate interval.

Learn how to evaluate the limits of a piecewise function as x approaches -2. Detailed step-by-step solution covering left-hand and right-hand limits. Suitable for advanced high school students and anyone studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation