*f*(*x*) = 

−5 − *x*

 + 6,

     if *x* < −6

6,

     if *x* = −6

2*x* + 19,

     if *x* > −6

Find each limit. (If the limit is infinite, enter '∞' or '-∞', as appropriate. If the limit does not otherwise exist, enter DNE.)

(a)

lim *x*→−6− *f*(*x*)
 
(b)

lim *x*→−6+ *f*(*x*)
 
(c)

lim *x*→−6 *f*(*x*)
 

To solve these problems, we will evaluate the piecewise function \( f(x) \) for different limits. The function \( f(x) \) is defined as:

\[
f(x) = 
\begin{cases} 
-5 - x + 6 & \text{if } x < -6 \\
6 & \text{if } x = -6 \\
2x + 19 & \text{if } x > -6 
\end{cases}
\]

First, let's simplify the function expressions:

- For \( x < -6 \): \( f(x) = -5 - x + 6 = 1 - x \)
- For \( x = -6 \): \( f(x) = 6 \)
- For \( x > -6 \): \( f(x) = 2x + 19 \)

Let's evaluate each limit:

### (a) \(\lim_{x \to -6^-} f(x)\)

This is the limit of \( f(x) \) as \( x \) approaches \(-6\) from the left. For \( x < -6 \), the function is defined as \( f(x) = 1 - x \).

So, as \( x \to -6 \) from the left:
\[
f(x) = 1 - x
\]

Plugging in \( x = -6 \):
\[
f(-6) = 1 - (-6) = 1 + 6 = 7
\]

Therefore:
\[
\lim_{x \to -6^-} f(x) = 7
\]

### (b) \(\lim_{x \to -6^+} f(x)\)

This is the limit of \( f(x) \) as \( x \) approaches \(-6\) from the right. For \( x > -6 \), the function is defined as \( f(x) = 2x + 19 \).

So, as \( x \to -6 \) from the right:
\[
f(x) = 2x + 19
\]

Plugging in \( x = -6 \):
\[
f(-6) = 2(-6) + 19 = -12 + 19 = 7
\]

Therefore:
\[
\lim_{x \to -6^+} f(x) = 7
\]

### (c) \(\lim_{x \to -6} f(x)\)

This is the two-sided limit of \( f(x) \) as \( x \) approaches \(-6\). The two-sided limit exists if the left-hand limit and the right-hand limit are equal.

From parts (a) and (b), we have:
\[
\lim_{x \to -6^-} f(x) = 7 \quad \text{and} \quad \lim_{x \to -6^+} f(x) = 7
\]

Since these two limits are equal, the two-sided limit exists and is equal to this value.

Therefore:
\[
\lim_{x \to -6} f(x) = 7
\]

### Summary of Answers

(a) \(\lim_{x \to -6^-} f(x) = 7\)  
(b) \(\lim_{x \to -6^+} f(x) = 7\)  
(c) \(\lim_{x \to -6} f(x) = 7\)  

Would you like more details on any part, or do you have any questions?

Here are some related questions you might consider:
1. What is \(f(-6)\)?
2. What would \(\lim_{x \to 0^-} f(x)\) be if the function had a different domain?
3. What are the continuity points of \(f(x)\)?
4. How do you graph this piecewise function?
5. Can you determine the derivatives of this piecewise function at different points?

**Tip:** When evaluating limits for piecewise functions, check the function definition for values approaching from both directions to determine if the limits match.

Learn how to evaluate limits of a piecewise function step-by-step. This problem involves determining limits approaching a specific point from different directions. Suitable for students in Grades 11-12 studying advanced calculus and piecewise functions.

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