One-Sided Limit of Real Function/Examples/Reciprocal of 1 + e to the Reciprocal of x
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Examples of One-Sided Limits of Real Functions
Let $f: \R \to \R$ be the real function defined as:
- $\map f x = \dfrac 1 {1 + e^{1 / x} }$
Then:
\(\ds \lim_{x \mathop \to 0^+} \map f x\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \lim_{x \mathop \to 0^-} \map f x\) | \(=\) | \(\ds 1\) |
Proof
We have:
- $\dfrac 1 x \to +\infty$ as $x \to 0^+$
and so:
- $1 + e^{1/x} \to +\infty$ as $x \to 0^+$
Thus:
- $\dfrac 1 {1 + e^{1 / x} } \to 0$ as $x \to 0^+$
and remains positive.
$\Box$
Then we have:
- $\dfrac 1 x \to -\infty$ as $x \to 0^-$
and so:
- $-e^{1/x} \to 0$ as $x \to 0^-$
Thus:
- $\dfrac 1 {1 + e^{1 / x} } \to 1$ as $x \to 0^-$
and remains less than $1$.
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.1$ Limits and Continuity: Example $\text C$