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A function $f$ is real analytic at $y$ when its Taylor seriesconverges to $f(x)$ throughout some neighborhood of $y$.

As $x\to 0$, the limit $\sin x/x\to 1$ gives $\sin x\sim x$, while the Taylor series gives $\arcsin x\sim x$ and $\cos x\sim 1-x^2/2$.

As $x\downarrow 0$, the exponential integral haswhere $\gamma$ is the Euler--Mascheroni constant. Integrating the Taylor series of $E_1^{\prime }(x)=-e^{-x}/x$ gives every nonconstant coefficient; the constant follows from the limiting definition of $\gamma$. A logarithm of a stretched variable can create a switchback term in a matched asymptotic expansion.

The taylor series at zero beginsso $\tanh x\sim x$ as $x\to 0$.

A Taylor expansion about a point consists of successive derivative terms there, together with either a finite remainder or an infinite Taylor series when that series represents the function.