Exponential Tends to Zero and Infinity

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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 is 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 $\openint 0 \infty$


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We have that the Exponential is Strictly Increasing.

From above, $\ds \lim_{x \mathop \to \infty} \exp x = \infty$

From above, $\ds \lim_{x \mathop \to -\infty} \exp x = 0$

Hence the result.

$\blacksquare$


Sources

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