Taylor Series

Let f(.) be a twice differentiable function from R² → R. If p, x₀ ∈ R² where ||p|| is sufficiently small (here ||. || is the Euclidean norm or distance function), then f(x₀ + p) =...

Let $f(.)$ be a twice differentiable function from $ \mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0} \in \mathbb{R}^{2}$ where $\vert \vert P\vert \vert$ is sufficien...

The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is

$$\mathop {Lt}\limits_{x \to 0} {{x - \sin x} \over {1 - \cos x}}$$ is

$$\,\mathop {Lim}\limits_{x \to 0} \left( {{{1 - \cos x} \over {{x^2}}}} \right)$$ is

In the Taylor series expansion of $${e^x}$$ about $$x=2,$$ the coefficient of $$\,\,{\left( {x - 2} \right)^4}\,\,$$ is

$$\mathop {Lim}\limits_{x \to 0} {{{e^x} - \left( {1 + x + {{{x^2}} \over 2}} \right)} \over {{x^3}}} = $$

$$\mathop {Lim}\limits_{x \to 0} {{x\left( {{e^x} - 1} \right) + 2\left( {\cos x - 1} \right)} \over {x\left( {1 - \cos x} \right)}} = \_\_\_\_\_\_.$$