Let's observe the convergence of the forward-difference formula applied to the function 
at 
.
at . f = @(x) sin( exp(x+1) );
FD1 = [ (f(0.1)-f(0)) /0.1
(f(0.05)-f(0)) /0.05
(f(0.025)-f(0)) /0.025 ]
It's not clear that the sequence is converging. As predicted, however, the errors are cut approximately by a factor of 2 when h is divided by 2.
exact_value = cos(exp(1))*exp(1);
err = exact_value - FD1
Asymptotically as 
, the error is proportional to h.
, the error is proportional to h.