Math Problem Statement
Is f(x) continuous at x = -1? Show the 3-step definition of continuity to justify your answer.
Solution
The problem asks if the function is continuous at and to use the 3-step definition of continuity to justify the answer.
The function is piecewise, defined as follows:
x + 3, & \text{if } x < -1 \\ 2, & \text{if } x = -1 \\ \frac{2}{1 + e^{1 + x}}, & \text{if } x > -1 \end{cases}$$ ### Steps for Checking Continuity at $$x = -1$$ To determine whether $$f(x)$$ is continuous at $$x = -1$$, we need to use the 3-step definition of continuity: 1. **Check if $$f(-1)$$ is defined**: We need to verify that the function has a value at $$x = -1$$. From the piecewise definition, $$f(-1) = 2$$. 2. **Check if the limit of $$f(x)$$ as $$x$$ approaches $$-1$$ from both sides exists**: - **Left-hand limit**: $$\lim_{x \to -1^-} f(x)$$ means $$x$$ approaches $$-1$$ from the left (i.e., $$x < -1$$). The expression for this is $$f(x) = x + 3$$. So: $$\lim_{x \to -1^-} f(x) = (-1) + 3 = 2$$ - **Right-hand limit**: $$\lim_{x \to -1^+} f(x)$$ means $$x$$ approaches $$-1$$ from the right (i.e., $$x > -1$$). The expression for this is $$f(x) = \frac{2}{1 + e^{1 + x}}$$. Let's compute this limit as $$x \to -1^+$$: $$\lim_{x \to -1^+} \frac{2}{1 + e^{1 + x}} = \frac{2}{1 + e^0} = \frac{2}{1 + 1} = 1$$ 3. **Check if the limit equals the function value at $$x = -1$$**: For the function to be continuous at $$x = -1$$, we need the left-hand limit, right-hand limit, and $$f(-1)$$ to all be equal. - Left-hand limit: 2 - Right-hand limit: 1 - $$f(-1) = 2$$ Since the right-hand limit does not equal the left-hand limit (2 ≠ 1), the function is **not continuous** at $$x = -1$$. Would you like more details on any step or further clarification? ### Related Questions: 1. What is the significance of left-hand and right-hand limits in continuity? 2. How do you compute the limit of an exponential function as $$x \to -1$$? 3. Can a function be continuous if only one of the one-sided limits equals the function value? 4. What types of functions are typically discontinuous at a point? 5. How does the definition of continuity extend to higher dimensions? **Tip**: When checking continuity at a point for piecewise functions, always carefully evaluate the limit from both sides.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Piecewise Functions
Formulas
lim (x -> -1^-) f(x) = x + 3
lim (x -> -1^+) f(x) = 2 / (1 + e^(1 + x))
Theorems
3-step definition of continuity
Suitable Grade Level
College-level Calculus