Let’s go through your solutions step by step to verify if they are correct.

Problem (a):

Evaluate
$\lim_{x \to \infty} \frac{2x - 1}{5x + 3}.$

Your work:

You rewrote the expression by dividing every term in the numerator and denominator by $x$: $\frac{2x - 1}{5x + 3} = \frac{\frac{2x}{x} - \frac{1}{x}}{\frac{5x}{x} + \frac{3}{x}} = \frac{\frac{2}{1} - \frac{1}{x}}{\frac{5}{1} + \frac{3}{x}}.$

As $x \to \infty$, the terms $-\frac{1}{x}$ and $\frac{3}{x}$ approach $0$. Thus: $\frac{\frac{2}{1} - 0}{\frac{5}{1} + 0} = \frac{2}{5}.$

Answer: Your solution is correct. The limit is: $\boxed{\frac{2}{5}}.$

Problem (b):

Evaluate
$\lim_{x \to \infty} \frac{\sqrt{x} - x^3}{2x + x^2}.$

Your work:

You factored $x^2$ from the numerator and denominator:

• Numerator: $\sqrt{x} - x^3 = x^3 \left(\frac{1}{x^{2.5}} - 1\right)$,
• Denominator: $2x + x^2 = x^2 \left(\frac{2}{x} + 1\right)$.

So the limit becomes: $\lim_{x \to \infty} \frac{\frac{1}{x^{2.5}} - 1}{\frac{2}{x} + 1}.$

As $x \to \infty$, $\frac{1}{x^{2.5}} \to 0$ and $\frac{2}{x} \to 0$, simplifying to: $\frac{-1}{1} = -\infty.$

Analysis: Your steps are incorrect.

Correct Solution:

Factor $x^3$ directly from the numerator and denominator. The numerator becomes: $\sqrt{x} - x^3 = x^3 \left(\frac{1}{x^{2.5}} - 1\right).$ The denominator becomes: $2x + x^2 = x^2 \left(\frac{2}{x} + 1\right).$

Now rewrite: $\frac{\sqrt{x} - x^3}{2x + x^2} = \frac{x^3 \left(\frac{1}{x^{2.5}} - 1\right)}{x^2 \left(\frac{2}{x} + 1\right)} = x \cdot \frac{\frac{1}{x^{2.5}} - 1}{\frac{2}{x} + 1}.$

As $x \to \infty$, the dominant term is $x \cdot \frac{-1}{1} = -x$. Thus: $\boxed{-\infty}.$

So your final result is correct, but the intermediate steps are unclear and inconsistent. It’s better to simplify directly.

Final Verdict:

1. (a) Correct.
2. (b) Correct final answer, but reasoning should be revised.

Would you like me to provide a detailed explanation or alternative approaches to these limits?

Follow-up Questions:

1. Why does dividing by the highest power of $x$ simplify these limits?
2. What happens if you try other methods (e.g., L'Hôpital's Rule) for these problems?
3. How can you confirm $-\infty$ as the behavior of the function in part (b)?
4. Can we analyze part (b) graphically to verify the behavior as $x \to \infty$?
5. What are key strategies for handling indeterminate forms like $\frac{\infty}{\infty}$?

Tip:

Always verify intermediate simplifications in limit problems to avoid errors in reasoning.