Question

find by applying L'hospital's Rule​

Answer 1

Answer:

              2/3

Step-by-step explanation:

L'hospital's Rule says that when we have something that approaches the form 0/0 (as we do here), then we can differentiate the numerator and denominator to get a new function with the same limit.  That is

• lim [ f(x) / g(x) ]  =  lim [ f'(x) / g'(x) ]

Here, the derivative of the numerator (x - sin x cos x) is

• d(x - sin x cos x) / dx  =  1 - cos 2x

and the derivative of the denominator x³ is

• d(x³) / dx  =  3x²

So we can instead evaluate the limit (as x → 0)

• lim (1 - cos 2x) / 3x²

But alas!  This also takes the form 0/0.  Not to worry, because this just means that we can use L'hospital's Rule again.

The derivative of the numerator (1 - cos 2x) is

• 2 sin 2x

and the derivative of the denominator 3x² is

• 6x

So we can instead evaluate the limit (as x → 0)

• lim (2 sin 2x) / 6x

This also takes the form 0/0 yet again.  We can either use L'hospital's Rule yet again to arrive at

• lim (2 sin 2x) / 6x  =  lim (4 cos 2x) / 6  =  4/6  =  2/3

or alternatively, we can use the familiar fact that lim (sin ax)/(ax) = 1, to get

• lim (2 sin 2x) / 6x  =  2/3 × lim (sin 2x)/2x  =  2/3 × 1  =  2/3

Hope that helps!