Math Problem Statement
Suppose that the function f(x) is continuous at x = -1 and that f(x) is defined by f(x) = { 4 - x^2, x < -1 ; ax^2 + 2, x >= -1 }. (a) Find the value of a to make f(x) continuous. (b) Evaluate lim_{x -> -1} f(x). (c) Evaluate lim_{x -> 0} f(x). (d) Evaluate lim_{x -> 2} f(x). (e) Evaluate lim_{x -> infinity} f(x).
Solution
The problem asks you to ensure that the function is continuous at and also involves evaluating various limits for . The piecewise function is defined as:
\begin{cases} 4 - x^2, & x < -1 \\ ax^2 + 2, & x \geq -1 \end{cases}$$ ### Part (a): Find $$a$$ to make $$f(x)$$ continuous at $$x = -1$$. To ensure continuity at $$x = -1$$, the left-hand limit and the right-hand limit at $$x = -1$$ must be equal, and this value must also be equal to $$f(-1)$$. 1. **Left-hand limit** as $$x \to -1^-$$ (using the first part of the function): $$\lim_{x \to -1^-} f(x) = 4 - (-1)^2 = 4 - 1 = 3$$ 2. **Right-hand limit** as $$x \to -1^+$$ (using the second part of the function): $$\lim_{x \to -1^+} f(x) = a(-1)^2 + 2 = a(1) + 2 = a + 2$$ For continuity, we require: $$3 = a + 2$$ Solving for $$a$$: $$a = 1$$ So, $$a = 1$$ makes the function continuous at $$x = -1$$. --- ### Part (b)-(e): Evaluate each limit. - **(b) $$\lim_{x \to -1} f(x)$$**: Since the function is continuous at $$x = -1$$, the limit as $$x \to -1$$ is simply $$f(-1)$$, which we computed as 3. So, $$\lim_{x \to -1} f(x) = 3$$ - **(c) $$\lim_{x \to 0} f(x)$$**: For $$x = 0$$, use the second piece $$f(x) = ax^2 + 2$$ with $$a = 1$$: $$\lim_{x \to 0} f(x) = (1)(0^2) + 2 = 2$$ - **(d) $$\lim_{x \to 2} f(x)$$**: For $$x = 2$$, use the second part $$f(x) = x^2 + 2$$: $$\lim_{x \to 2} f(x) = (1)(2^2) + 2 = 4 + 2 = 6$$ - **(e) $$\lim_{x \to \infty} f(x)$$**: For $$x \to \infty$$, the second part of the function dominates: $$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} (x^2 + 2) = \infty$$ Would you like more details on any part of this problem? --- Here are 5 related questions to expand understanding: 1. What is the formal definition of continuity at a point? 2. Why do we require both the left-hand and right-hand limits to match for continuity? 3. How would the function change if $$a \neq 1$$? 4. What would happen if the function had a different behavior for $$x > -1$$? 5. How do you approach limits at infinity for different types of functions? ### Tip: Always check both the left-hand and right-hand limits to determine continuity at a point, especially for piecewise functions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
f(x) = 4 - x^2 for x < -1
f(x) = ax^2 + 2 for x >= -1
Theorems
Definition of Continuity
Limit Laws
Suitable Grade Level
Grade 12 - Undergraduate Calculus