determine whether f is continuous at the given point F(x) = x^2+4/ x-2 when x+3

The function provided is \( f(x) = \frac{x^2 + 4}{x - 2} \), and we are tasked with determining whether it is continuous at \( x = 3 \).

To check continuity at a given point, \( x = c \), the following conditions must be met:

1. \( f(c) \) is defined.
2. The limit \( \lim_{x \to c} f(x) \) exists.
3. The limit equals the function value: \( \lim_{x \to c} f(x) = f(c) \).

### Step 1: Check if \( f(3) \) is defined

Substitute \( x = 3 \) into the function:
\[
f(3) = \frac{3^2 + 4}{3 - 2} = \frac{9 + 4}{1} = \frac{13}{1} = 13
\]
So, \( f(3) \) is defined and equals 13.

### Step 2: Check if \( \lim_{x \to 3} f(x) \) exists

To compute the limit, simply substitute \( x = 3 \) into the expression:
\[
\lim_{x \to 3} f(x) = \lim_{x \to 3} \frac{x^2 + 4}{x - 2} = \frac{3^2 + 4}{3 - 2} = \frac{9 + 4}{1} = 13
\]
The limit exists and equals 13.

### Step 3: Check if the limit equals \( f(3) \)

Since both the limit and the function value at \( x = 3 \) are 13, the third condition for continuity is satisfied.

### Conclusion:
The function \( f(x) \) is continuous at \( x = 3 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What happens if \( f(x) \) is evaluated at \( x = 2 \)?
2. How do we handle discontinuities in rational functions?
3. Can a function have a limit at a point but not be continuous at that point?
4. What are different types of discontinuities?
5. How can we determine if a piecewise function is continuous at a given point?

**Tip:** Always check for undefined points in rational functions, as they often cause discontinuities.

determine whether f is continuous at the given point F(x) = x^2+4/ x-2 when x=3

Explore how to determine the continuity of the function f(x) = (x^2 + 4) / (x - 2) at x = 3. This problem involves key concepts like continuity and limits in rational functions, making it suitable for Grades 10-12. Follow the detailed step-by-step solution to understand how to evaluate the function and the limit at a given point.

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