Limit of Real Function/Examples/e^-1 over size x at 0

Example of Limit of Real Function

 * $\ds \lim_{x \mathop \to 0} e^{-1 / \size x} = 0$

Proof

 * Limit-of-e-to-minus-1-over-size x.png

By definition of the limit of a real function:
 * $\ds \lim_{x \mathop \to a} \map f x = A$


 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - a} < \delta \implies \size {\map f x - A} < \epsilon$
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - a} < \delta \implies \size {\map f x - A} < \epsilon$

Let $\epsilon \in \R_{>0}$ be chosen arbitrarily.

Then we have:

So, having been given an arbitrary $\epsilon \in \R_{>0}$, let $\delta = \dfrac 1 {\map \ln {1 / \epsilon} }$.

Then:
 * $0 < \size x < \delta \implies \size {e^{-1 / \size x} } < \epsilon$

Hence by definition of limit of a real function:
 * $\ds \lim_{x \mathop \to 0} e^{-1 / \size x} = 0$