Math Problem Statement
Find each point of discontinuity of f(x), and for each point of discontinuity, evaluate the indicated one-sided limits. Given f(x) as: f(x) = 7 for x < 6, f(x) = 0 for x = 6, and f(x) = 10 + x for x > 6.
Solution
Let's analyze the given piecewise function :
\begin{cases} 7, & x < 6, \\ 0, & x = 6, \\ 10 + x, & x > 6. \end{cases}$$ We need to find the points of discontinuity and evaluate the one-sided limits at each of these points. ### Step 1: Check continuity at $$x = 6$$. For $$f(x)$$ to be continuous at $$x = 6$$, the following conditions must be satisfied: 1. $$\lim_{x \to 6^-} f(x) = \lim_{x \to 6^+} f(x)$$ (the left-hand and right-hand limits must exist and be equal). 2. The value of the function at $$x = 6$$ must match the limits: $$\lim_{x \to 6} f(x) = f(6)$$. #### 1. Left-hand limit as $$x \to 6^-$$: When $$x < 6$$, $$f(x) = 7$$. $$\lim_{x \to 6^-} f(x) = 7.$$ #### 2. Right-hand limit as $$x \to 6^+$$: When $$x > 6$$, $$f(x) = 10 + x$$. $$\lim_{x \to 6^+} f(x) = 10 + 6 = 16.$$ #### 3. Value of the function at $$x = 6$$: $$f(6) = 0$$. Since $$\lim_{x \to 6^-} f(x) = 7$$ and $$\lim_{x \to 6^+} f(x) = 16$$, these one-sided limits are not equal. Hence, $$f(x)$$ is **discontinuous** at $$x = 6$$. Additionally, $$f(6) = 0$$, which does not match either of the limits. ### Conclusion: - $$x = 6$$ is a point of discontinuity. - The one-sided limits are: - $$\lim_{x \to 6^-} f(x) = 7$$, - $$\lim_{x \to 6^+} f(x) = 16$$. Would you like a deeper explanation on this or any follow-up questions? Here are some relative questions to expand on the topic: 1. How do you formally define the concept of continuity at a point? 2. Can you explain the difference between jump discontinuity and removable discontinuity? 3. What are other types of discontinuities besides jump discontinuity? 4. How do you handle discontinuities in real-world applications? 5. What are some common techniques to evaluate limits involving piecewise functions? **Tip**: When dealing with piecewise functions, always check the one-sided limits at the points where the definition of the function changes to determine discontinuities.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Continuity
One-Sided Limits
Discontinuities
Formulas
-
Theorems
Continuity at a Point
Limit Definition
Suitable Grade Level
Grades 10-12