Calculus Question

Question

Evaluate the following limit and describe the result of this limit.

 

cos(x+h)-cosx/(h) as h approaches 0. help!?

Answer 1

Recall:

{1} cos ( A + B ) = cos(A)cos(B) - sin(A)sin(B)

{2} lim_h->0 [ (cos(h) - 1 )/h ] = 0

{3} lim_h->0 [sin(h) / h ] = 1

 

lim_h->0 [ ( cos(x+h) - cos(x) ) / h ]     **use {1}

 

lim_h->0 [ ( cos(x)cos(h) - sin(x)sin(h) - cosx ) / h ]

 

lim_h->0 [ (cos(x)cos(h) - cosx - sin(x)sin(h)  ) / h ]

 

lim_h->0 [ ( cos(x) (cos(h) - 1 ) - sin(x)sin(h)  ) / h ]

 

lim_h->0 [ ( cos(x) (cos(h) -1) )/h  -  ( sin(x)sin(h) ) /h ]

 

lim_h->0 [ cos(x) · (cos(h) - 1)/h ]  -  lim_h->0 [ sin(x) · sin(h)/h ]  ** use {2} and {3}

 

cos(x) · (0) - sin(x) · (1)

 

- sin(x)