Let's evaluate the limit expression you've provided:

$\lim_{x \to \infty} \frac{3x^3 + 5x - 1}{x^2 - 3x + a}$

Step-by-Step Solution:

1. Analyze the degrees of the numerator and denominator:

• The numerator is $3x^3 + 5x - 1$. The highest degree term here is $3x^3$.
• The denominator is $x^2 - 3x + a$. The highest degree term here is $x^2$.

As $x \to \infty$, the highest-degree terms will dominate in both the numerator and the denominator.

• Numerator: $3x^3$
• Denominator: $x^2$

2. Simplify the expression for large $x$:

For large $x$, the lower-degree terms become negligible compared to the highest-degree terms. Therefore, we can approximate the original expression as:

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

3. Take the limit:

Now, as $x \to \infty$, the term $3x$ tends to infinity.

Thus:

$\lim_{x \to \infty} \frac{3x^3 + 5x - 1}{x^2 - 3x + a} = \infty$

Conclusion:

The limit does not exist in a finite sense because the expression tends to infinity as $x \to \infty$.

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Would you like a more detailed explanation on any part, or have any further questions on limits?

Related Questions:

1. How does the degree of the highest term affect the limit of rational functions?
2. What happens when the degrees of the numerator and denominator are equal in a limit expression?
3. How can I handle limits of rational functions with both degrees greater than 1?
4. What if the highest degree terms in the numerator and denominator have the same power?
5. Can you explain limits involving indeterminate forms like $\frac{\infty}{\infty}$?

Tip: Always focus on the highest-degree terms in both the numerator and denominator when taking limits as $x \to \infty$. These terms generally determine the behavior of the function.