lim x tending to 0 (((cos x)^(1/2) - (cos x)^(1/3)) / (sin x)^2).Please solve this

To solve the limit as \( x \) approaches 0 for the expression \(\frac{(\cos x)^{1/2} - (\cos x)^{1/3}}{(\sin x)^2}\), we can start by analyzing the behavior of both the numerator and the denominator as \( x \) approaches 0.

Step 1: Simplifying the Expression

First, let's rewrite the expression for clarity:

\[ \lim_{x \to 0} \frac{(\cos x)^{1/2} - (\cos x)^{1/3}}{(\sin x)^2} \]

Step 2: Evaluating the Denominator

As \( x \) approaches 0, we know that \(\sin x\) behaves like \(x\). Therefore, \((\sin x)^2\) approaches \(0\) as well:

\[ \sin x \approx x \implies (\sin x)^2 \approx x^2 \]

Step 3: Evaluating the Numerator

Next, we need to analyze the numerator \((\cos x)^{1/2} - (\cos x)^{1/3}\). As \( x \) approaches 0, \(\cos x\) approaches 1. Thus, we can use the Taylor series expansion for \(\cos x\) around 0:

\[ \cos x \approx 1 - \frac{x^2}{2} + O(x^4) \]

Substituting this into our expression:

\[ (\cos x)^{1/2} \approx \left(1 - \frac{x^2}{2}\right)^{1/2} \quad \text{and} \quad (\cos x)^{1/3} \approx \left(1 - \frac{x^2}{2}\right)^{1/3} \]

Step 4: Using Binomial Expansion

We can apply the binomial expansion for small values of \( x \):

• \((1 + u)^n \approx 1 + nu\) for small \( u \).

For \((\cos x)^{1/2}\):

\[ (\cos x)^{1/2} \approx 1 - \frac{1}{4}\left(\frac{x^2}{2}\right) = 1 - \frac{x^2}{8} \]

For \((\cos x)^{1/3}\):

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

Step 5: Finding the Difference

Now, we can find the difference:

\[ (\cos x)^{1/2} - (\cos x)^{1/3} \approx \left(1 - \frac{x^2}{8}\right) - \left(1 - \frac{x^2}{6}\right) = \frac{x^2}{6} - \frac{x^2}{8} \]

To combine these fractions, we find a common denominator (24):

\[ \frac{x^2}{6} - \frac{x^2}{8} = \frac{4x^2}{24} - \frac{3x^2}{24} = \frac{x^2}{24} \]

Step 6: Substituting Back into the Limit

Now we substitute this back into our limit:

\[ \lim_{x \to 0} \frac{\frac{x^2}{24}}{(\sin x)^2} \approx \lim_{x \to 0} \frac{\frac{x^2}{24}}{x^2} = \lim_{x \to 0} \frac{1}{24} = \frac{1}{24} \]

Final Result

Thus, the limit as \( x \) approaches 0 for the given expression is:

\[ \frac{1}{24} \]