if x=0 what is the answer of :(TAN(X)-SIN(X))/X^3?

Question

an expert is needed
i want the limit

Answer 1

If you want the LIMIT, that's different than saying "x=0". Because x=0 gives you indeterminate form (0/0).

But since it does give you indeterminiate form, you can use l'Hopital's rule, which says the limit should be the same as the derivative of the top over the derivative of the bottom. This gives you (sec²(x) - cos(x)) / (3x²). Plugging in zero again gives you (1-1)/0 = 0/0, so once again you can use l'Hopital's rule. The next derivative is (2sec(x)(sec(x)tan(x)) + sin(x))/ 6x, or (2sec²(x)tan(x) + sin(x))/6x. When x=0, this is (0 + 0)/0 = 0/0, so use l'Hopital's rule again. (We have to keep checking that the value gives 0/0, otherwise we can't apply the rule).

Taking the derivative one more time, we get
(4sec(x)(sec(x)tan(x))*tan(x) + 2sec²(x)*(sec²(x)) + cos(x)) / 6

Plug in 0 and you get
(4*1*1*0*0 + 2(1)(1) + 1) / 6 = (0+3)/6 = 1/2. So the limit is 1/2

Answer 2

The limit as x --> 0 from the left or the right is 1/2.

Answer 3

it is 0/0
take derivative of numerator and denominar 3 times

the limit is 3/4