We are tasked with evaluating the following limit:

$\lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x}.$

Let us break this problem into steps:

Step 1: Approximation of $\cos x$

Near $x = 0$, we use the Taylor series approximation: $\cos x \approx 1 - \frac{x^2}{2}.$

Step 2: Expand $\sqrt{\cos x}$ and $\sqrt[3]{\cos x}$

Using binomial expansions:

• For $\sqrt{\cos x} = (\cos x)^{1/2}$: $\sqrt{\cos x} \approx \sqrt{1 - \frac{x^2}{2}} \approx 1 - \frac{x^2}{4}.$

• For $\sqrt[3]{\cos x} = (\cos x)^{1/3}$: [ \sqrt[3]{\cos x} \approx \sqrt[3]{1 - \frac{x^2}{2}} \approx 1 - \frac{x^2}{6}. ]

Step 3: Simplify the numerator

$\sqrt{\cos x} - \sqrt[3]{\cos x} \approx \left(1 - \frac{x^2}{4}\right) - \left(1 - \frac{x^2}{6}\right) = \frac{x^2}{6} - \frac{x^2}{4}.$

Simplify further: $\sqrt{\cos x} - \sqrt[3]{\cos x} \approx \frac{x^2}{12}.$

Step 4: Approximation of $\sin^2 x$

Near $x = 0$, $\sin x \approx x$. Thus: $\sin^2 x \approx x^2.$

Step 5: Simplify the fraction

Substitute into the limit: $\frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x} \approx \frac{\frac{x^2}{12}}{x^2}.$

Simplify: $\frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x} \approx \frac{1}{12}.$

Step 6: Evaluate the limit

As $x \to 0$, the limit is: $\lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x} = \frac{1}{12}.$

Final Answer:

$\boxed{\frac{1}{12}}$

Would you like a detailed explanation of any specific step? Here are some related questions to expand understanding:

1. How do Taylor series expansions help in solving limits involving trigonometric functions?
2. Can the same technique be applied to other similar problems involving higher-order roots?
3. What is the justification for approximating $\sin x \approx x$ near zero?
4. How does the binomial theorem aid in expanding functions near their base values?
5. Could L’Hôpital’s Rule be used to evaluate this limit instead?

Tip: When solving trigonometric limits, remember that Taylor approximations and simplifications near small angles ($x \to 0$) are often the most efficient methods