https://file.aiquickdraw.com/mathbot/file/1727053625598e1mcflce.jpg

Let's break down the problem step by step:

### Function Definition
The given function \( f(x) \) is defined as:

\[
f(x) =
\begin{cases}
\frac{|x - 7|}{x^2 - 49} & \text{if } x \neq 7 \\
\frac{1}{x + a} & \text{if } x = 7
\end{cases}
\]

We need to answer the following parts based on this function.

---

### Part (a): Simplify for \( x < 7 \)
For \( x < 7 \), we know that:

\[
|x - 7| = -(x - 7) = 7 - x
\]

So, the expression becomes:

\[
\frac{|x - 7|}{x^2 - 49} = \frac{7 - x}{x^2 - 49}
\]

Now, factor \( x^2 - 49 \):

\[
x^2 - 49 = (x - 7)(x + 7)
\]

Thus:

\[
\frac{7 - x}{x^2 - 49} = \frac{7 - x}{(x - 7)(x + 7)} = \frac{-(x - 7)}{(x - 7)(x + 7)} = \frac{-1}{x + 7} \quad \text{(for \( x \neq 7 \))}
\]

---

### Part (b): Compute the limit as \( x \to 7^- \)
The simplified expression as \( x \to 7^- \) is:

\[
\frac{-1}{x + 7}
\]

Taking the limit as \( x \to 7^- \):

\[
\lim_{x \to 7^-} \frac{-1}{x + 7} = \frac{-1}{7 + 7} = \frac{-1}{14}
\]

---

### Part (c): Simplify for \( x > 7 \)
For \( x > 7 \), we know that:

\[
|x - 7| = x - 7
\]

So, the expression becomes:

\[
\frac{|x - 7|}{x^2 - 49} = \frac{x - 7}{x^2 - 49}
\]

Again, factor \( x^2 - 49 \):

\[
x^2 - 49 = (x - 7)(x + 7)
\]

Thus:

\[
\frac{x - 7}{x^2 - 49} = \frac{x - 7}{(x - 7)(x + 7)} = \frac{1}{x + 7} \quad \text{(for \( x \neq 7 \))}
\]

---

### Part (d): Compute the limit as \( x \to 7^+ \)
The simplified expression as \( x \to 7^+ \) is:

\[
\frac{1}{x + 7}
\]

Taking the limit as \( x \to 7^+ \):

\[
\lim_{x \to 7^+} \frac{1}{x + 7} = \frac{1}{7 + 7} = \frac{1}{14}
\]

---

### Part (e): Find \( a \) such that \( f(x) \) is continuous at \( x = 7 \)
For \( f(x) \) to be continuous at \( x = 7 \), we need:

\[
\lim_{x \to 7} f(x) = f(7)
\]

From parts (b) and (d), we have:

\[
\lim_{x \to 7^-} f(x) = \frac{-1}{14}, \quad \lim_{x \to 7^+} f(x) = \frac{1}{14}
\]

For continuity, these limits must equal \( f(7) \), i.e., \( \frac{1}{7 + a} \). However, since the left- and right-hand limits are not equal, it is not possible to find an \( a \) that makes \( f(x) \) continuous at \( x = 7 \).

Thus, the answer is **empty**.

---

Would you like any more details on these steps? Here are some related questions for further exploration:

1. How do we determine if a function is continuous at a point?
2. What other conditions besides continuity might require limits to match?
3. Why does factoring the denominator help in simplifying expressions like these?
4. How do you handle limits approaching from both sides of a point?
5. Could the function be redefined to be continuous at \( x = 7 \) in some other way?

**Tip:** Always check if left- and right-hand limits match when determining continuity at a point.

Consider the function f(x) defined as f(x) = |x - 7| / (x^2 - 49) for x ≠ 7, and 1 / (x + a) for x = 7. Solve the following: a) Simplify the expression for x < 7 without using absolute value. b) Compute the limit as x approaches 7 from the left. c) Simplify the expression for x > 7 without using absolute value. d) Compute the limit as x approaches 7 from the right. e) Find the value of a such that the function is continuous at x = 7.

Learn how to solve a piecewise function involving absolute values and limits, simplifying expressions, and finding the value of a for continuity at x = 7. This problem involves factoring, handling limits from both sides, and applying theorems of continuity. Suitable for advanced high school or university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation