Question

find the lim x->0 of xsinx/1-cosx

Answer 1

use L'hospital's rule

Answer 2

but i dont know how to derive the trigonometric functions

Answer 3

do you know the derivative of sinx and cosx?

Answer 4

the numerator is a product function, so you would use the product rule

Answer 5

hint
the derivative of sinx = cosx
the derivative of cosx = -sinx

Answer 6

and the derivate of xsinx?

Answer 7

use the product rule \[\huge \color{red}{f'(x)}g(x) + f(x)\color{red}{g'(x)}\]

Answer 8

sinx+xcosx this is the derivative of xsinx?

Answer 9

yeah

Answer 10

the expression now is sinx+xcosx/sinx

Answer 11

what should i do now?

Answer 12

plug in zero to x and see what you get
if the limit in undefined then, use use L'hospital's rule again

Answer 13

cosx-xsinx/cosx?
and the limit is 1?

Answer 14

what is the derivative of sinx+xcosx
and dont forget the product rule on xcosx

Answer 15

the derivative of sinx+xcosx= cosx-xsinx
the derivative of sinx+xcosx/sinx=cosx-xsinx/cosx
so the limit should be 1?

Answer 16

the derivative of sinx+xcosx = cosx + cosx - xsinx, simplifies to = 2cosx-xsinx