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Have some more that I do not understand.
Limit theta approaching zero
SinTheta minus Tan theta/ sin cubed theta
Rewrite as [sin x(1 - 1/cosx)]/sin^3x
= [1 - (1/cosx)]/[1 - cos^2x]
Now use L'Hopital's rule and take the ratio of the derivatives of numerator and denominator. That lets you get rid of the 1's.
You are left with
Lim [-tanx secx/(2cosx sinx)]
= Lim -1/(2 cos^3x)
Which is -1/2
It took me a while to get that one. I cheated and used a hand calculator first, and it agrees.
To evaluate the limit of the expression as theta approaches zero, we can use L'Hôpital's rule. This rule allows us to find the limit of a fraction of two functions by taking the derivatives of both the numerator and the denominator and then evaluating the limit again.
Let's start by finding the derivative of the numerator and denominator separately:
1. The derivative of sin(theta) is cos(theta).
2. The derivative of tan(theta) is sec^2(theta).
3. The derivative of sin^3(theta) is 3sin^2(theta)cos(theta).
Now, let's rewrite the expression with these derivatives:
lim(theta → 0) [cos(theta) - sec^2(theta)] / [3sin^2(theta)cos(theta)].
Next, substitute theta = 0 into the expression:
[cos(0) - sec^2(0)] / [3sin^2(0)cos(0)].
Since cos(0) = 1 and sin(0) = 0, this simplifies to:
[1 - sec^2(0)] / [0 * cos(0)].
Since sec(0) is equal to 1, this further simplifies to:
[1 - 1] / [0 * 1].
Finally, [0/0] is an indeterminate form, indicating that further simplification is necessary. We can apply L'Hôpital's rule once more:
Taking the derivatives of the numerator and denominator:
The derivative of [1 - sec^2(0)] is 0 (since sec^2(0) is a constant).
The derivative of [0 * cos(0)] is 0 (since both terms in the product are constants).
So, we get:
lim(theta → 0) 0/ 0.
This is still an indeterminate form. We can continue applying L'Hôpital's rule until we get a definite result.
Taking the derivatives again:
The derivative of 0 is 0.
The derivative of 0 is 0.
This means that we have reached a result of 0/0 again, which is still indeterminate.
At this point, we can try another approach to solving the limit. By applying trigonometric identities, we can rewrite the expression as follows:
lim(theta → 0) [sin(theta) - (sin(theta)/cos(theta))] / [sin^3(theta)].
Simplifying, we get:
lim(theta → 0) [sin(theta)(cos(theta) - 1)] / [sin^3(theta)].
Now, we can substitute theta = 0 into the expression:
0(1) / 0^3.
This results in 0/0 once again.
Since we have exhausted the possibilities using L'Hôpital's rule and simplification, we can conclude that the limit does not exist for this particular expression as theta approaches zero.