Exponential Tends to Zero and Infinity
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
The structure of this page is truly inconsistent with house style. I am also a little concerned that arguments in this area may be circular. One needs to tread very carefully in this area.
It may also need to be brought up to our standard house style.
If you have done this or know it has been done, please remove {{Tidy}} from the code.
Theorem
Let $x \in \R$ be a real number.
Let $\exp x$ be the exponential of $x$.
Then:
- $\exp x \to +\infty$ as $x \to +\infty$
- $\exp x \to 0$ as $x \to -\infty$
Thus the exponential function has domain $\R$ and image $\openint 0 \infty$.
Proof
The exponential function approaches positive infinity as x approaches positive infinity
Let $M$ be a strictly positive real number.
Let $N$ be $\ln M$.
$N$ is real because $M > 0$.
From Exponential is Strictly Increasing:
- $\forall x: x > N \implies \exp x > \exp N = M$
Therefore:
- $\forall M \in \R_{>0} : \exists N \in \R : \forall x > N : \exp x > M$
The result follows from the definition of infinite limit at infinity.
$\blacksquare$
The exponential function approaches 0 as x approaches negative infinity
Let $\epsilon$ be a strictly positive real number.
Let $c$ be $\ln \epsilon$.
Let $x$ be any real number that smaller than $c$.
From Exponential of Real Number is Strictly Positive:
- $\exp x > 0 \implies \size {\exp x} = \exp x$
From Exponential is Strictly Increasing:
- $\size {\exp x - 0} = \exp x < \exp c = \epsilon$
Therefore:
- $\forall \epsilon \in \R_{>0} : \exists c \in \R : \forall x < c : \size {\exp x - 0} < \epsilon$
From the definition of limit at minus infinity, the result follows.
$\blacksquare$
The exponential function has domain $\R$ and image $\openint 0 \infty$
the above uses this result
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by resolving the issues.
To discuss this proof in more detail, feel free to use the talk page.
If you are able to fix this page, then when you have done so you can remove this instance of {{Questionable}} from the code.
If you would welcome a second opinion as to whether your correction is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
We have that the Exponential is Strictly Increasing.
From above, $\displaystyle \lim_{x \mathop \to \infty} \exp x = \infty$
From above, $\displaystyle \lim_{x \mathop \to -\infty} \exp x = 0$
Hence the result.
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.4$