Math Problem Statement
Consider the following function:
f(x) =
(6/x, if x < 0; x^2 - 3x, if 0 <= x < 2; 5x, if x >= 2)
Find the following limits:
1. lim(x -> 0-) f(x)
2. lim(x -> 0+) f(x)
3. lim(x -> 1) f(x)
4. Identify the type of discontinuity at x = 2, if any.
Solution
Let's analyze the piecewise function given in the problem:
\begin{cases} \frac{6}{x}, & \text{if } x < 0 \\ x^2 - 3x, & \text{if } 0 \leq x < 2 \\ 5x, & \text{if } x \geq 2 \end{cases}$$ We are tasked with finding the following limits: --- ### a. $$\lim_{x \to 0^-} f(x)$$ For $$x \to 0^-$$, we use the function defined for $$x < 0$$, which is $$f(x) = \frac{6}{x}$$. $$\lim_{x \to 0^-} \frac{6}{x} = -\infty$$ --- ### b. $$\lim_{x \to 0^+} f(x)$$ For $$x \to 0^+$$, we use the function defined for $$0 \leq x < 2$$, which is $$f(x) = x^2 - 3x$$. $$f(0^+) = 0^2 - 3(0) = 0$$ $$\lim_{x \to 0^+} f(x) = 0$$ --- ### c. $$\lim_{x \to 1} f(x)$$ For $$0 \leq x < 2$$, the function is $$f(x) = x^2 - 3x$$, so we evaluate: $$f(1) = 1^2 - 3(1) = 1 - 3 = -2$$ Thus: $$\lim_{x \to 1} f(x) = -2$$ --- ### d. Discontinuity at $$x = 2$$? We need to check whether the function is continuous at $$x = 2$$. - Left-hand limit ($$x \to 2^-$$) is $$f(x) = x^2 - 3x$$. Substituting $$x = 2$$: $$f(2^-) = 2^2 - 3(2) = 4 - 6 = -2$$ - Right-hand limit ($$x \to 2^+$$) is $$f(x) = 5x$$. Substituting $$x = 2$$: $$f(2^+) = 5(2) = 10$$ Since the left-hand and right-hand limits are not equal, there is a **jump discontinuity** at $$x = 2$$. --- ### Summary of answers: - a. $$-\infty$$ - b. $$0$$ - c. $$-2$$ - d. A jump discontinuity --- Would you like more details on any of these calculations? Here are five related questions to explore: 1. What are the types of discontinuities, and how do they differ? 2. How can limits be evaluated graphically for piecewise functions? 3. What is the significance of one-sided limits in piecewise functions? 4. How do you check continuity for a piecewise function at a given point? 5. How can limits involving infinity be interpreted in real-world contexts? **Tip**: When evaluating limits of piecewise functions, always ensure you're using the correct part of the function for the given range.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Piecewise Functions
Formulas
Limit of a function
Discontinuity Types
Theorems
Definition of limits
Types of discontinuities
Suitable Grade Level
Undergraduate or Advanced High School Calculus