One-Sided Limit of Real Function/Examples

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Examples of One-Sided Limits of Real Functions

Example: $\dfrac 1 {1 + e^{1 / x} }$

Let $f: \R \to \R$ be the real function defined as:

$\map f x = \dfrac 1 {1 + e^{1 / x} }$

Then:

\(\ds \lim_{x \mathop \to 0^+} \map f x\) \(=\) \(\ds 0\)
\(\ds \lim_{x \mathop \to 0^-} \map f x\) \(=\) \(\ds 1\)


Example: $e^{-1 / x}$ to $0$ from Right

Let $f: \R \to \R$ be the real function defined as:

$\map f x = e^{-1 / x}$

Then:

\(\ds \lim_{x \mathop \to 0^+} \map f x\) \(=\) \(\ds 0\)


Example: $e^{-1 / x}$ to $0$ from Left

Let $f: \R \to \R$ be the real function defined as:

$\map f x = e^{-1 / x}$

Then:

\(\ds \lim_{x \mathop \to 0^-} \map f x\) \(=\) \(\ds +\infty\)
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