determine if $$ f\left(x\right)=\frac{2x^2+3x+1}{x^2+5x} $$ is continuous at a=5

To determine if the function $ f(x) = \frac{2x^2 + 3x + 1}{x^2 + 5x} $ is continuous at $ a = 5 $, we must check three conditions for continuity at that point:

1. $ f(a) $ is defined (the function exists at $ a = 5 $),
2. The limit $ \lim_{x \to a} f(x) $ exists,
3. The value of the function at $ a = 5 $ is equal to the limit as $ x $ approaches 5.

### Step 1: Check if $ f(5) $ is defined
To check if $ f(x) $ is defined at $ a = 5 $, substitute $ x = 5 $ into the function:

\[
f(5) = \frac{2(5)^2 + 3(5) + 1}{(5)^2 + 5(5)} = \frac{2(25) + 15 + 1}{25 + 25} = \frac{50 + 15 + 1}{50} = \frac{66}{50} = 1.32
\]

Thus, $ f(5) $ is defined, and its value is $ 1.32 $.

### Step 2: Check if $ \lim_{x \to 5} f(x) $ exists
Now, compute the limit as $ x $ approaches 5. Since $ f(x) $ is a rational function and the denominator does not become zero when $ x = 5 $, we can directly substitute $ x = 5 $ into the limit expression:

\[
\lim_{x \to 5} f(x) = \frac{2(5)^2 + 3(5) + 1}{(5)^2 + 5(5)} = \frac{66}{50} = 1.32
\]

The limit exists and is equal to $ 1.32 $.

### Step 3: Check if $ f(5) = \lim_{x \to 5} f(x) $
We have already found that:
- $ f(5) = 1.32 $
- $ \lim_{x \to 5} f(x) = 1.32 $

Since both the value of the function and the limit at $ x = 5 $ are equal, the function is continuous at $ a = 5 $.

### Final Conclusion:
The function $ f(x) = \frac{2x^2 + 3x + 1}{x^2 + 5x} $ is continuous at $ a = 5 $.

Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. How do you determine continuity for piecewise functions?
2. What happens if the denominator of a rational function is zero at a point?
3. How do you check the continuity of a function with absolute values?
4. What are the key differences between continuity and differentiability?
5. How do you check continuity at a point for trigonometric functions?
6. How can you graphically interpret the continuity of a function?
7. How do limits behave when discontinuities like holes or jumps are present?
8. What role does continuity play in the Intermediate Value Theorem?

**Tip:** Continuity at a point requires that the function be defined at the point, the limit exists, and the value of the function equals the limit.

Learn how to determine the continuity of the function f(x) = (2x^2 + 3x + 1) / (x^2 + 5x) at x = 5. Understand the conditions required for continuity, including the existence of the function at the point, the existence of the limit, and the equality of the function value and the limit. Suitable for advanced high school students in Grades 11-12.

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