determine if $$ f\left(x\right)=\frac{5x-2}{x^2-9x+20} $$ is continuous at a=4

To determine if the function $ f(x) = \frac{5x - 2}{x^2 - 9x + 20} $ is continuous at $ a = 4 $, we follow the three conditions for continuity at a point:

1. $ f(a) $ is defined,
2. $ \lim_{x \to a} f(x) $ exists,
3. The value of the function at $ a = 4 $ is equal to the limit as $ x $ approaches 4.

### Step 1: Check if $ f(4) $ is defined
Substitute $ x = 4 $ into the function:

- **Numerator**: $ 5(4) - 2 = 20 - 2 = 18 $,
- **Denominator**: $ 4^2 - 9(4) + 20 = 16 - 36 + 20 = 0 $.

Since the denominator is zero at $ x = 4 $, $ f(4) $ is not defined.

### Conclusion:
Because the function $ f(x) $ is not defined at $ a = 4 $, the function is **not continuous** at that point.

Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. How do you handle discontinuities caused by division by zero?
2. What is a removable discontinuity, and how do you identify it?
3. How do you check continuity at a point for a piecewise function?
4. How do you determine the type of discontinuity in a rational function?
5. How do you apply limits to check for continuity in a function with undefined points?
6. How do holes in the graph relate to discontinuities?
7. How can you rewrite functions to remove removable discontinuities?
8. What is the difference between jump discontinuities and removable discontinuities?

**Tip:** Always check if the denominator of a rational function becomes zero at a given point. If it does, the function may not be defined, leading to a discontinuity.

Explore how to determine the continuity of a rational function f(x) = (5x - 2) / (x^2 - 9x + 20) at a = 4. Understand the conditions required for continuity, including the existence of f(a) and the limit of f(x) as x approaches a. Learn about discontinuities caused by division by zero and their implications in rational functions.

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