To find the limit

$\lim_{x \to 0^+} \frac{(\cos(x))^3}{x^2},$

we first substitute $x = 0$:

$\frac{(\cos(0))^3}{0^2} = \frac{1^3}{0} = \frac{1}{0},$

which indicates that the limit approaches infinity. However, to confirm whether the limit diverges or has a specific behavior, we can analyze the function further.

As $x$ approaches $0^+$, $\cos(x)$ approaches $1$. Therefore, $(\cos(x))^3$ approaches $1$ as well. The denominator $x^2$ approaches $0$. Thus, we have:

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

To use l'Hôpital's Rule, we can re-examine the limit. Since both the numerator and denominator approach $0$ as $x \to 0^+$, we can rewrite it as follows:

1. Apply l'Hôpital's Rule:

We differentiate the numerator and denominator:

$\text{Numerator: } \frac{d}{dx} \left((\cos(x))^3\right) = 3(\cos(x))^2 (-\sin(x)) = -3(\cos(x))^2 \sin(x),$

$\text{Denominator: } \frac{d}{dx}(x^2) = 2x.$

Thus, we have:

$\lim_{x \to 0^+} \frac{(\cos(x))^3}{x^2} = \lim_{x \to 0^+} \frac{-3(\cos(x))^2 \sin(x)}{2x}.$

Now substituting $x = 0$:

The numerator approaches $0$ (since $\sin(0) = 0$) and the denominator approaches $0$ as well, allowing us to apply l'Hôpital's Rule again.

2. Apply l'Hôpital's Rule a second time:

Differentiate again:

• The numerator becomes:

$\frac{d}{dx} \left(-3(\cos(x))^2 \sin(x)\right) = -3 \left(2\cos(x)(-\sin(x)) \sin(x) + (\cos(x))^2 \cos(x)\right) = 3 \left(2\cos(x)\sin^2(x) - \cos^3(x)\right),$

• The denominator becomes:

$\frac{d}{dx}(2x) = 2.$

Now the limit is:

$\lim_{x \to 0^+} \frac{3(2\cos(x)\sin^2(x) - \cos^3(x))}{2}.$

As $x \to 0^+$, this evaluates to:

$\lim_{x \to 0^+} \frac{3(2 \cdot 1 \cdot 0 - 1)}{2} = \frac{3(-1)}{2} = -\frac{3}{2}.$

However, since we want the limit of the original function as $x \to 0^+$, we find:

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

Thus, the final answer is:

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

Summary:

The limit diverges to positive infinity.

Do you want further details on this solution or have any questions? Here are some related questions you might consider:

1. What is the behavior of $\cos(x)$ as $x$ approaches other values?
2. How does l'Hôpital's Rule apply to different types of indeterminate forms?
3. Can you explain the Taylor series expansion for $\cos(x)$?
4. What are other limits involving trigonometric functions?
5. How does the limit change if we consider $\lim_{x \to 0} \frac{(\cos(x))^3}{x^2}$ from both sides?

Tip: Always check the form of the limit before deciding on the method to apply!