f[x_] := Cos[x]

Plot[f[x], {x, 0, 0.5}]

Limit[f[x], x -> 0]

data =
Table[
Table[
{1/2^i, N[f[1/2^ i]]},
{i, 1, n}
],
{n, 1, 10}
];

TableForm[data]

int[i_, h_] := InterpolatingPolynomial[data[[i]], h] // N

Table[
int[i, x] // Simplify,
{i, 1, 4}
];
TableForm[%]

points = Table[
{i, Re[int[i, 0]]},
{i, 1, 10}
]

ListPlot[points]