What is the derivative of -3sin(θ)cos(θ) ? (It's not 6 sin2(θ) . )whatlifeforme said:Homework Statement
use l'hopital's to evaluate the limit.
Homework Equations
Code:limit (∅->0) ∅-3sin∅cos ∅ -------------------- tan∅- ∅The Attempt at a Solution
i take the derivatives of the top and bottom, and use trig identity for sec^2x - 1 to tan^x.
i get 6(cosx)^2 annd get the value of 6. but the answer is -∞.
6(sinx)^2
----------
(tanx)^2
6(cosx)^2
SammyS said:What is the derivative of -3sin(θ)cos(θ) ? (It's not 6 sin2(θ) . )
SammyS said:What is the derivative of -3sin(θ)cos(θ) ? (It's not 6 sin2(θ) . )
Yes, using the product rule, [itex]\displaystyle \ \ \frac{d}{dx}\left(\sin(xcos(x)\right)=\cos^2(x)-\sin^2(x)\ \ [/itex] and that's equal to [itex]\ \cos(2x)\ .\ [/itex]whatlifeforme said:using the product rule:
-3( (cosx)^2 - (sinx)^2 ) == cos2x
which means:
1-3(cos2x)
------------
(tanx)^2
further:
1-(3/2)sin(2x)
-----------------
2tanx * (secx)^2
i'm still not getting to -infinity, and it seems to be getting complicated in the denominator.
The L'Hopital's rule is a mathematical technique used to evaluate limits involving trigonometric functions. It states that if the limit of a function is indeterminate (such as 0/0 or infinity/infinity), then the limit of the quotient of the derivatives of the numerator and denominator will be the same as the limit of the original function.
L'Hopital's rule should be used when the limit of a trigonometric function cannot be solved using other methods, such as algebraic manipulation or substitution. It is also useful when the function involves trigonometric identities that make it difficult to evaluate the limit directly.
Yes, L'Hopital's rule can be applied to any trigonometric function, whether it is a single trigonometric function or a combination of multiple trigonometric functions. It can also be used for limits involving inverse trigonometric functions.
One limitation of L'Hopital's rule for trigonometric functions is that it can only be used for limits that approach either 0 or infinity. It cannot be used for limits that approach a constant value. Additionally, the function must be differentiable at the point where the limit is being evaluated.
To apply L'Hopital's rule for trigonometric limits, first determine if the limit is indeterminate (such as 0/0 or infinity/infinity). Then, take the derivative of both the numerator and denominator of the function. Finally, evaluate the limit of the quotient of the derivatives. If the original limit is still indeterminate, the process can be repeated until the limit is no longer indeterminate.