https://file.aiquickdraw.com/mathbot/file/1726517873116dtbyu6x4.jpg

To determine whether the function \( f(x) \) has a jump discontinuity at \( x = 7 \), we need to compute the left-hand and right-hand limits of \( f(x) \) as \( x \) approaches 7.

The function is defined as:
\[
f(x) = 
\begin{cases}
5x - 5 & \text{if } x < 7 \\
4 & \text{if } x \geq 7
\end{cases}
\]

### Step 1: Compute the left-hand limit \( \lim_{x \to 7^-} f(x) \)

For \( x < 7 \), we use the piece \( f(x) = 5x - 5 \). So,
\[
\lim_{x \to 7^-} f(x) = \lim_{x \to 7^-} (5x - 5) = 5(7) - 5 = 35 - 5 = 30
\]

### Step 2: Compute the right-hand limit \( \lim_{x \to 7^+} f(x) \)

For \( x \geq 7 \), we use the piece \( f(x) = 4 \). So,
\[
\lim_{x \to 7^+} f(x) = 4
\]

### Step 3: Compare the limits

We have:
\[
\lim_{x \to 7^-} f(x) = 30 \quad \text{and} \quad \lim_{x \to 7^+} f(x) = 4
\]

Since these limits are not equal, \( f(x) \) has a jump discontinuity at \( x = 7 \).

Would you like further clarification on any part of the solution?

### Related questions:
1. What are the types of discontinuities other than jump discontinuities?
2. How can we graphically interpret a jump discontinuity?
3. How does the definition of a piecewise function affect continuity?
4. Can a function have more than one type of discontinuity at a single point?
5. What happens if the left-hand and right-hand limits both exist but are equal?

**Tip:** A function can have a jump discontinuity even if the limits exist, as long as they are not equal.

Show that f(x) has a jump discontinuity at x = 7 by calculating the limits from the left and right at x = 7.

Learn how to prove a jump discontinuity in a piecewise function f(x) at x = 7 by calculating left-hand and right-hand limits. This problem involves limits, piecewise functions, and continuity concepts. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation