Math Problem Statement
Estudiar la continuidad de la función f(x) en los puntos x=1 y x=2, donde f(x) está definida como una función a trozos con la siguiente expresión: f(x) = x*L(x) / (x^2 - 1) si x ≠ 1 y f(x) = 1/2 si x = 1.
Solution
The image shows a piecewise-defined function and a question about its continuity at certain points, specifically and .
The function is defined as:
\begin{cases} \frac{x \cdot L(x)}{x^2 - 1}, & \text{if } x \neq 1 \\ \frac{1}{2}, & \text{if } x = 1 \end{cases}$$ Where $$L(x)$$ is not explicitly defined in the image, but we will assume it is some continuous function. ### Steps to check the continuity at $$x = 1$$: 1. **Limit at $$x = 1$$ from both sides**: - The first step is to calculate $$\lim_{x \to 1} f(x)$$ for the case when $$x \neq 1$$, meaning we use the expression $$\frac{x \cdot L(x)}{x^2 - 1}$$. - Notice that $$x^2 - 1 = (x - 1)(x + 1)$$, so the function will have a removable discontinuity if the numerator cancels out the denominator when $$x = 1$$. 2. **Value of the function at $$x = 1$$**: - The function is explicitly defined as $$f(1) = \frac{1}{2}$$. For the function to be continuous at $$x = 1$$, the limit $$\lim_{x \to 1} f(x)$$ must equal $$f(1) = \frac{1}{2}$$. Would you like a detailed breakdown of how to calculate this limit, or do you have more specific questions regarding this function? ### Related Questions: 1. How do you compute limits for piecewise-defined functions? 2. What is the difference between continuity and differentiability? 3. What happens if the limit does not exist at a specific point? 4. Can a function be continuous but not differentiable? If so, how? 5. How do removable discontinuities affect the graph of a function? #### Tip: When solving for limits involving indeterminate forms (like $$0/0$$), always consider factoring or using L'Hôpital's Rule to simplify the expressions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Continuity
Piecewise Functions
Limits
Formulas
f(x) = x * L(x) / (x^2 - 1) for x ≠ 1
Limit definition of continuity: lim(x→c) f(x) = f(c)
Theorems
Limit and Continuity Theorem
Factorization of Quadratic Expressions
Suitable Grade Level
Grades 11-12
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