The image shows two limit problems. Let's solve them step by step.

(a) $\lim_{x \to 0} \frac{x - \sin(x)}{x^3}$

Step 1: Apply L'Hopital's Rule

This limit is an indeterminate form $\frac{0}{0}$ as both the numerator $x - \sin(x)$ and the denominator $x^3$ approach 0 as $x \to 0$. We can apply L'Hopital's Rule, which states that if the limit is of the form $\frac{0}{0}$, we can take the derivatives of the numerator and the denominator.

• Derivative of the numerator: $\frac{d}{dx}(x - \sin(x)) = 1 - \cos(x)$
• Derivative of the denominator: $\frac{d}{dx}(x^3) = 3x^2$

Thus, the limit becomes: $\lim_{x \to 0} \frac{1 - \cos(x)}{3x^2}$

Step 2: Apply L'Hopital's Rule Again

This is still of the form $\frac{0}{0}$, so we apply L'Hopital's Rule again:

• Derivative of the numerator: $\frac{d}{dx}(1 - \cos(x)) = \sin(x)$
• Derivative of the denominator: $\frac{d}{dx}(3x^2) = 6x$

Now the limit becomes: $\lim_{x \to 0} \frac{\sin(x)}{6x}$

Step 3: Solve the Limit

We know that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$, so the expression simplifies to: $\frac{1}{6}$

Thus, the solution to (a) is: $\boxed{\frac{1}{6}}$

(b) $\lim_{x \to \infty} \frac{x^3 \cos(x^2)}{3x^4 + 2}$

Step 1: Analyze the Degree of Terms

As $x \to \infty$, the denominator grows as $3x^4$, which dominates over the constant term 2. In the numerator, we have $x^3 \cos(x^2)$. The cosine function oscillates between -1 and 1, meaning the numerator's magnitude is bounded by $-x^3$ and $x^3$.

Step 2: Estimate the Limit

We can think of $\cos(x^2)$ as bounded while $x^3$ grows large. So, the numerator behaves like $x^3$, and the denominator behaves like $3x^4$. This suggests that:

$\lim_{x \to \infty} \frac{x^3 \cos(x^2)}{3x^4 + 2} \approx \lim_{x \to \infty} \frac{x^3}{3x^4} = \lim_{x \to \infty} \frac{1}{3x} = 0$

Thus, the solution to (b) is: $\boxed{0}$

Would you like further details or clarifications? Here are five additional related questions for further exploration:

1. How is L'Hopital's Rule used in cases involving $\frac{0}{0}$ indeterminate forms?
2. Can L'Hopital's Rule be applied when the limit is of the form $\frac{\infty}{\infty}$?
3. How would you evaluate a limit if the denominator grows much faster than the numerator?
4. What are other techniques for solving limits involving trigonometric functions besides L'Hopital's Rule?
5. How do the bounds on $\cos(x)$ help simplify limits when it oscillates?

Tip: When applying L'Hopital's Rule, make sure the limit is in an indeterminate form before differentiating the numerator and denominator.