Math Problem Statement
Определить непрерывность или тип разрыва функции f(x) в точке x_0, где f(x) = \left(1 + x)^{\frac{1}{x}} для x < 0 и f(x) = e для x \geq 0 при x_0 = 0.
Solution
To analyze the continuity of the function at , let's go through the definition and types of discontinuity.
The function is given as:
(1 + x)^{\frac{1}{x}}, & x < 0 \\ e, & x \geq 0 \end{cases}$$ ### Step-by-Step Solution 1. **Right-hand Limit (as $$x \to 0^+$$)**: Since $$f(x) = e$$ for $$x \geq 0$$, the right-hand limit of $$f(x)$$ as $$x$$ approaches 0 from the right is: $$\lim_{x \to 0^+} f(x) = e$$ 2. **Left-hand Limit (as $$x \to 0^-$$)**: For $$x < 0$$, $$f(x) = (1 + x)^{\frac{1}{x}}$$. The limit of this expression as $$x \to 0^-$$ is also known to converge to $$e$$: $$\lim_{x \to 0^-} f(x) = e$$ 3. **Value at $$x = 0$$**: The function defines $$f(0) = e$$, so: $$f(0) = e$$ ### Conclusion Since the left-hand limit, right-hand limit, and the function's value at $$x = 0$$ all equal $$e$$, the function $$f(x)$$ is **continuous** at $$x = 0$$. ### Answer The correct choice is: - **Непрерывность** (continuity). Would you like further details or have additional questions? --- Here are 5 additional questions related to continuity and types of discontinuity: 1. What are the different types of discontinuities, and how are they classified? 2. How do we calculate limits involving indeterminate forms, like $$(1 + x)^{1/x}$$ as $$x \to 0^-$$? 3. What is the difference between removable and non-removable discontinuities? 4. How do we determine if a piecewise function is continuous at a boundary point? 5. How does the continuity of a function relate to differentiability? **Tip:** When dealing with limits, especially near points where expressions are complex, using L'Hôpital's rule or known limits can simplify the process.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Limits
Types of Discontinuities
Formulas
Left-hand limit \lim_{x \to 0^-} f(x)
Right-hand limit \lim_{x \to 0^+} f(x)
Function value f(0)
Theorems
Continuity at a Point
Properties of Limits
Suitable Grade Level
University Level