https://file.aiquickdraw.com/mathbot/file/1728390301326l8wtoc5p.jpg

The image shows a piecewise-defined function \( f(x) \) and a question about its continuity at certain points, specifically \( x = 1 \) and \( x = 2 \).

The function is defined as:

\[
f(x) =
\begin{cases}
\frac{x \cdot L(x)}{x^2 - 1}, & \text{if } x \neq 1 \\
\frac{1}{2}, & \text{if } x = 1
\end{cases}
\]

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.

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.

Learn how to study the continuity of a piecewise-defined function at specific points, x=1 and x=2. This problem involves calculating limits, understanding the behavior of the function, and using key algebraic concepts such as factorization and continuity definitions. Suitable for advanced high school students in Grades 11-12.

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