For a function f, let
f*(x)=lim (f(x+h)-f(x-h))/h
h->0
Determine f*(x) for f(x)=cos(x)
cos(x+h)=cosxcosh-sinxsinh
cos(x-h)=cosxcosh+sinxsinh
subtracting
-2sinx sinh/h=-2sinx
To determine f*(x) for f(x) = cos(x), we need to compute the limit of (f(x+h) - f(x-h))/h as h approaches 0.
Step 1: Substitute f(x) = cos(x) into the expression for f*(x):
f*(x) = lim (cos(x+h) - cos(x-h))/h, as h->0
Step 2: Apply the limit definition and simplify:
f*(x) = lim [(cos(x)cos(h) - sin(x)sin(h)) - (cos(x)cos(-h) - sin(x)sin(-h))]/h, as h->0
= lim [(cos(x)cos(h) - sin(x)sin(h)) - (cos(x)cos(h) + sin(x)sin(h))]/h, as h->0 (using the identity: cos(-h) = cos(h) and sin(-h) = -sin(h))
= lim [-2sin(x)sin(h)]/h, as h->0 (simplifying the expression)
Step 3: Apply L'Hôpital's Rule to evaluate the limit:
f*(x) = lim [-2sin(x)cos(h)]/1, as h->0 (applying L'Hôpital's Rule)
= -2sin(x) (evaluating the limit as h approaches 0)
Therefore, f*(x) = -2sin(x) for f(x) = cos(x).