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determine if f is continous at the given values

To determine if the function \( f(x) \) is continuous at the given values, \( x = -1 \) and \( x = 10 \), let's analyze the piecewise function provided in the image:

\[
f(x) = 
\begin{cases} 
\frac{x^2 + 5x + 4}{x^2 - 3x + 2} & \text{for } x \neq -1 \\
3 & \text{for } x = -1 
\end{cases}
\]

### 1. Continuity at \( x = -1 \):

For the function to be continuous at \( x = -1 \), the following must hold:
\[
\lim_{x \to -1} f(x) = f(-1)
\]

First, we calculate the limit as \( x \) approaches \(-1\) using the first case of the function:

\[
\lim_{x \to -1} \frac{x^2 + 5x + 4}{x^2 - 3x + 2}
\]

We simplify the expression inside the limit:

- The numerator \( x^2 + 5x + 4 \) can be factored as \( (x + 4)(x + 1) \).
- The denominator \( x^2 - 3x + 2 \) can be factored as \( (x - 1)(x - 2) \).

So the expression becomes:

\[
\frac{(x + 4)(x + 1)}{(x - 1)(x - 2)}
\]

Evaluating the limit as \( x \) approaches \(-1\):

\[
\lim_{x \to -1} \frac{(x + 4)(x + 1)}{(x - 1)(x - 2)} = \frac{(-1 + 4)(-1 + 1)}{(-1 - 1)(-1 - 2)} = \frac{3 \times 0}{(-2) \times (-3)} = 0
\]

Now, \( f(-1) = 3 \) (from the second case of the function).

Since \( \lim_{x \to -1} f(x) = 0 \) and \( f(-1) = 3 \), the function is **not continuous** at \( x = -1 \).

### 2. Continuity at \( x = 10 \):

To check for continuity at \( x = 10 \), we only need to ensure that the limit and the function value at \( x = 10 \) are equal because \( x = 10 \) falls under the first case of the piecewise function.

\[
f(10) = \frac{10^2 + 5(10) + 4}{10^2 - 3(10) + 2} = \frac{100 + 50 + 4}{100 - 30 + 2} = \frac{154}{72} = \frac{77}{36}
\]

Since there is no discontinuity in the expression at \( x = 10 \), the function is **continuous** at \( x = 10 \).

### Conclusion:

- **At \( x = -1 \):** The function is **not continuous**.
- **At \( x = 10 \):** The function is **continuous**.

Would you like more details or have any questions?

Here are some related questions:

1. What does it mean for a function to be continuous at a point?
2. How do you factor quadratic expressions like \( x^2 + 5x + 4 \)?
3. What is the significance of the limit in determining continuity?
4. How do you calculate limits involving rational functions?
5. What are common types of discontinuities in functions?

**Tip:** Always check the limit from both sides when determining continuity at a specific point, especially for piecewise functions.

Determine if the function f(x) is continuous at the given values x = -1 and x = 10.

Analyze the continuity of a piecewise function at x = -1 and x = 10. Learn how to apply limits and the definition of continuity to rational and piecewise functions. Suitable for Grades 11-12 students studying advanced algebra or calculus.

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