Limit of (1-cos^3x)/(x sinx cosx) as x approaches 0

To find the limit of the given expression as x approaches 0, we can simplify the expression and use the concept of L'Hôpital's rule, which allows us to evaluate limits involving indeterminate forms.

Simplifying the expression:
Let's simplify the given expression step by step:

1. Use the identity cos^2x = 1 - sin^2x to rewrite the numerator:
1 - cos^3x = 1 - (cos^2x)(cosx) = 1 - (1 - sin^2x)(cosx) = 1 - (cosx - sin^2x cosx)

2. Distribute the negative sign in the expression:
1 - (cosx - sin^2x cosx) = 1 - cosx + sin^2x cosx

3. Now, let's rewrite the denominator:
x sinx cosx = (sinx/x) * (cosx/x) = sinx/x * cosx/x = (sinx/x)^2

4. Substitute the simplified numerator and denominator back into the original expression:
(1 - cosx + sin^2x cosx) / (sinx/x)^2

Applying L'Hôpital's rule:
Now, we can apply L'Hôpital's rule to evaluate the limit of the simplified expression. According to the rule, if the limit of the ratio of derivatives of two functions exists, then the limit of the original function also exists and is equal to the limit of the ratio of their derivatives.

1. Take the derivative of the numerator and denominator separately:
- Derivative of 1 - cosx + sin^2x cosx = sinx + 2sinxcosx - sin^2x
- Derivative of (sinx/x)^2 = 2sinx/x^3

2. Substitute the derivatives back into the original expression:
limx→0 (sinx + 2sinxcosx - sin^2x) / (2sinx/x^3)

3. Now, let's evaluate the limit as x approaches 0:
At x = 0, the numerator and denominator both become 0. Hence, we have an indeterminate form.

4. Apply L'Hôpital's rule again by taking the derivatives of the numerator and denominator:
- Derivative of sinx + 2sinxcosx - sin^2x = cosx + 2cos^2x - 2sinx cosx
- Derivative of (2sinx/x^3) = (2cosx - 6sinx/x^4)

5. Substitute the derivatives back into the expression:
limx→0 (cosx + 2cos^2x - 2sinx cosx) / (2cosx - 6sinx/x^4)

6. Evaluate the limit as x approaches 0:
At x = 0, the numerator and denominator both become 1. Therefore, the limit is 1.

Conclusion:
The limit of (1-cos^3x)/(x sinx cosx) as x approaches 0 is equal to

x (1-cosx)(1+cos^2x+cosx)/×^2sinxcosx(1-cosx)/x^2=1/2x/sinx=11+cos^2x+cosx=3than 3×1×1/2=3/2