Math Problem Statement
Show that f(x) has a jump discontinuity at x = 7 by calculating the limits from the left and right at x = 7.
Solution
To determine whether the function has a jump discontinuity at , we need to compute the left-hand and right-hand limits of as approaches 7.
The function is defined as:
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Functions
Jump Discontinuity
Formulas
lim_{x \to a^-} f(x) = L
lim_{x \to a^+} f(x) = R
Jump discontinuity if L ≠ R
Theorems
Limit Definition
Piecewise Function Theorem
Continuity Theorem
Suitable Grade Level
Grades 11-12