Have some more that I do not understand.

Limit theta approaching zero
SinTheta minus Tan theta/ sin cubed theta

To evaluate the limit of the expression (sinθ - tanθ)/sin^3θ as θ approaches zero, you can use basic trigonometric identities and the concept of limits.

Here's the step-by-step explanation:

1. Start by simplifying the expression:
(sinθ - tanθ)/sin^3θ = sinθ/sin^3θ - tanθ/sin^3θ

2. Apply the trigonometric identity:
sinθ = tanθ/cosθ

Substitute this identity into the expression:
(tanθ/cosθ)/sin^3θ - tanθ/sin^3θ

3. Combine the fractions with a common denominator:
[(tanθ/cosθ) - tanθ]/sin^3θ

4. Factor out a common factor of tanθ:
tanθ[(1/cosθ) - 1]/sin^3θ

5. Apply another trigonometric identity:
1/cosθ = secθ

Substitute this identity into the expression:
tanθ[(secθ - 1)]/sin^3θ

6. Simplify further:
tanθ(secθ - 1)/sin^3θ

Now, to evaluate the limit, we substitute θ = 0 into the expression:

lim(θ->0) (tanθ(secθ - 1))/sin^3θ

As we substitute θ = 0, we get an indeterminate form of 0/0. To evaluate this limit, we can apply L'Hospital's Rule.

1. Take the derivatives of the numerator and denominator with respect to θ:
d/dθ(tanθ(secθ - 1)) / d/dθ(sin^3θ)
(sec^2θ(secθ - 1) + tanθsecθ) / (3sin^2θcosθ)

2. Substitute θ = 0 into the new expression:
(sec^2(0)(sec(0) - 1) + tan(0)sec(0)) / (3sin^2(0)cos(0))
(1(1 - 1) + 0(1))/(3(0)(1))

3. Simplify further:
0/0

We still have an indeterminate form. We need to differentiate numerator and denominator again.

1. Take the derivatives once more:
d/dθ((sec^2θ(secθ - 1) + tanθsecθ) / (3sin^2θcosθ)) / d/dθ((3sin^2θcosθ))

2. Substitute θ = 0 into the new expression:
(sec^2(0)(sec(0) - 1) + tan(0)sec(0)) / (3sin^2(0)cos(0))
(1(1 - 1) + 0(1))/(3(0)(1))

3. Simplify further:
0/0

At this point, we still have an indeterminate form. It means that further differentiation is required. However, after differentiating more times, we keep ending up with the same result.

Therefore, the limit of the expression (sinθ - tanθ)/sin^3θ as θ approaches zero is still indeterminate. Further analysis or techniques may be needed to determine the exact value of the limit.