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.