Limit at Infinity of Identity Function

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Theorem

Let $I_\R: \R \to \R$ be the identity function on $\R$.

Then:

$\displaystyle \lim_{x \to +\infty} \ I_\R \left({x}\right) = +\infty$
$\displaystyle \lim_{x \to -\infty} \ I_\R \left({x}\right) = -\infty$


Proof

We have that the Derivative of Identity Function is $1$.

Hence, by Derivative of Monotone Function, $x$ is strictly increasing.

Now, by the definition of infinite limits at infinity, the first assertion is:

$\forall M \in \R_{>0}: \exists N \in \R_{>0}: x > N \implies f \left({x}\right) > M$

For every $M$, choose $N = M$.

The second assertion is proved similarly.

$\blacksquare$