We are tasked with evaluating the limit:

$\lim_{x \to \infty} \left( \frac{5 + 3x}{x^3} - \frac{3 - x}{x + 7} + \frac{4 + e^{-x} + x^3}{2x^3 + 17x^2 + 1} \right)$

To solve this limit, let's evaluate each term individually as $x \to \infty$.

First Term: $\frac{5 + 3x}{x^3}$

As $x \to \infty$, this term can be simplified by dividing both the numerator and denominator by $x^3$:

$\frac{5 + 3x}{x^3} = \frac{5}{x^3} + \frac{3x}{x^3} = \frac{5}{x^3} + \frac{3}{x^2}$

Since $\frac{5}{x^3} \to 0$ and $\frac{3}{x^2} \to 0$ as $x \to \infty$, we get:

$\lim_{x \to \infty} \frac{5 + 3x}{x^3} = 0$

Second Term: $\frac{3 - x}{x + 7}$

As $x \to \infty$, the dominant term in both the numerator and denominator is $x$. We can approximate this term as:

$\frac{3 - x}{x + 7} \approx \frac{-x}{x} = -1$

Thus, the limit of this term is:

$\lim_{x \to \infty} \frac{3 - x}{x + 7} = -1$

Third Term: $\frac{4 + e^{-x} + x^3}{2x^3 + 17x^2 + 1}$

As $x \to \infty$, the dominant term in the numerator is $x^3$, and the dominant term in the denominator is also $x^3$. Therefore, we can simplify this term as:

$\frac{4 + e^{-x} + x^3}{2x^3 + 17x^2 + 1} \approx \frac{x^3}{2x^3} = \frac{1}{2}$

Thus, the limit of this term is:

$\lim_{x \to \infty} \frac{4 + e^{-x} + x^3}{2x^3 + 17x^2 + 1} = \frac{1}{2}$

Combining the Results

Now, we combine the limits of the three terms:

$0 - 1 + \frac{1}{2} = -\frac{1}{2}$

Thus, the final answer is:

$\lim_{x \to \infty} \left( \frac{5 + 3x}{x^3} - \frac{3 - x}{x + 7} + \frac{4 + e^{-x} + x^3}{2x^3 + 17x^2 + 1} \right) = -\frac{1}{2}$

Would you like more details on any step?

Here are 5 related questions to further your understanding:

1. How do we determine which terms are dominant as $x \to \infty$?
2. What is the importance of simplifying terms when calculating limits?
3. How does the behavior of exponential terms like $e^{-x}$ affect limits at infinity?
4. Can L'Hôpital's Rule be applied in cases like this? Why or why not?
5. How do rational functions behave as $x \to \infty$?

Tip: When working with limits at infinity, always compare the growth rates of the highest-degree terms in both the numerator and denominator.