The image shows a mathematical problem involving the identification of a jump discontinuity in a piecewise function $f(x)$ at $x = 5$. Here's a step-by-step approach to address this.

Problem Breakdown:

1. Definition of a Jump Discontinuity: A function $f(x)$ has a jump discontinuity at $x = a$ if:

   • $\lim_{x \to a^-} f(x)$ exists,
   • $\lim_{x \to a^+} f(x)$ exists, and
   • The left and right limits are not equal.

2. Piecewise Function Definition:

   \begin{cases} x^2 + 3x + 7 & \text{if } x < 5, \\ 7 & \text{if } x = 5, \\ -2x + 4 & \text{if } x > 5. \end{cases}$$

3. Task: We need to show that $f(x)$ has a jump discontinuity at $x = 5$ by calculating the left-hand limit $\lim_{x \to 5^-} f(x)$ and the right-hand limit $\lim_{x \to 5^+} f(x)$.

Step-by-Step Solution:

1. Left-hand Limit ($x \to 5^-$):

For $x < 5$, $f(x) = x^2 + 3x + 7$. $\lim_{x \to 5^-} f(x) = \lim_{x \to 5^-} (x^2 + 3x + 7).$ Substitute $x = 5$ into the expression: $f(5^-) = 5^2 + 3(5) + 7 = 25 + 15 + 7 = 47.$

2. Right-hand Limit ($x \to 5^+$):

For $x > 5$, $f(x) = -2x + 4$. $\lim_{x \to 5^+} f(x) = \lim_{x \to 5^+} (-2x + 4).$ Substitute $x = 5$ into the expression: $f(5^+) = -2(5) + 4 = -10 + 4 = -6.$

3. Check for Jump Discontinuity:

Since $\lim_{x \to 5^-} f(x) = 47$ and $\lim_{x \to 5^+} f(x) = -6$, and these two limits are not equal, there is a jump discontinuity at $x = 5$.

Graphing $f(x)$:

The graph would show different behaviors for $x < 5$, $x = 5$, and $x > 5$:

• For $x < 5$, the graph follows the parabola $x^2 + 3x + 7$.
• At $x = 5$, the function takes a constant value $7$.
• For $x > 5$, the graph follows the line $-2x + 4$.

Would you like more details or any specific explanations on this problem?

Here are 5 related questions you can explore:

1. What are the conditions for a function to have a removable discontinuity?
2. How do you classify other types of discontinuities (e.g., infinite discontinuity)?
3. What is the difference between a jump discontinuity and a removable discontinuity?
4. How can you extend this piecewise function to make it continuous at $x = 5$?
5. What is the significance of limits when analyzing the behavior of piecewise functions?

Tip: When evaluating limits for piecewise functions, always make sure to consider the specific domain for each piece of the function carefully.