Question #57a66

1 Answer
Feb 10, 2018

#b) f(x)=cos(x),c=pi/6#

Explanation:

We know:
#cos(pi/6)=sqrt3/2#

This means we can rewrite the limit like so:
#lim_(h->0)(cos(pi/6+h)-cos(pi/6))/h#

Considering the definition of a derivative of a function #f(x)# at a point #c#:
#lim_(h->0)(f(c+h)-f(c))/h#

A reasonable guess is that #c=pi \/ 6#, and using it, we can see that the inputs to the cosine function match up with the inputs to #f(x)# in the definition:
#lim_(h->0)(cos(color(red)(c+h))-cos(color(red)(c)))/h#

This means that if #c=pi \/ 6#, then #f(x)=cos(x)#.