Limits? ?The following limits represent the derivative of a function f at a point a. Find a possible f and a, and then evaluate the limit. 2 ? 1 lim sin (4+h)?2 h?0 h

Solution Step 1: sin (4+h)2 Given lim h h0 2 From the given limit the possible function f=sin x and the point a = 4 sin 4 +h)2 Then after evaluating the limit lim h directly we get 0/0 form h0 So we use L'Hospital's rule L’Hospital’s Rule tells us that if we have an indeterminate form 0/0 or all we need to do is differentiate the numerator and differentiate the denominator and then take the limit. sin 4 +h)2 2sin 4 +h)cos4( +h)0 lim h =lim 1 h0 h0 =lim 2sin ( +h4cos ( +h) 4 h0 =2sin ( )cos ( ) 4 4 =2 1 1 (sin ( ) = 1 , cos ( ) = 1 ) 2 2 4 2 4 2 = 2 2 =1 Therefore 2 1 lim sin 4 +h)2 =1 h0 h