Limit Properties

Arithmetic with Limits

Theorem: Arithmetic with Real Limits

Let $f,g:D\subseteq \mathbb{R}\to \mathbb{R}$ be real functions.

If both limits $\underset{x\to c}{\lim }f(x)$ and $\underset{x\to c}{\lim }g(x)$ exist for $c\in D\cup \{-\infty ,+\infty \}$, then

$$\begin{array}{rl}& \underset{x\to c}{\lim }\alpha f(x)+\beta g(x)=\alpha \underset{x\to c}{\lim }f(x)+\beta \underset{x\to c}{\lim }g(x)\forall \alpha ,\beta \in \mathbb{R}\\ & \\ & \underset{x\to c}{\lim }\left(f(x)g(x)\right)=\left(\underset{x\to c}{\lim }f(x)\right)\cdot \left(\underset{x\to c}{\lim }g(x)\right)\\ & \\ & \underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\frac{\underset{x\to c}{\lim }f(x)}{\underset{x\to c}{\lim }g(x)},\underset{x\to c}{\lim }g(x)\ne 0\end{array}$$

PROOF

TODO

WARNING

These do not apply to infinite limits.

Theorem: Arithmetic with Infinite Limits

Let $f,g:D\subseteq \mathbb{R}\to \mathbb{R}$ be two functions.

The following rules apply for the limits of $f$ and $g$ for $c\in D\cup \{-\infty ,+\infty \}$, no matter if they are real or infinite:

	$\underset{x\to c}{\lim }f(x)=L<0$	$\underset{x\to c}{\lim }f(x)=L>0$
$\underset{x\to c}{\lim }g(x)=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=-\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=-\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$
$\underset{x\to c}{\lim }g(x)=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=+\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=+\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$

	$\underset{x\to c}{\lim }f(x)=-\infty$	$\underset{x\to c}{\lim }f(x)=+\infty$
$\underset{x\to c}{\lim }g(x)=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=-\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=?$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$
$\underset{x\to c}{\lim }g(x)=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=?$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=+\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$

NOTE

A question mark (”?”) indicates that we cannot compute the limit directly, but we can try to transform the expression via algebraic manipulations in such a way, so as to make the limit computable.

PROOF

TODO

Squeeze Theorem

Theorem: The Squeeze Theorem for Functions

Let $f,g,h:D\subseteq \mathbb{R}\to \mathbb{R}$ be real functions.

If the limit of $f$ and $g$ as $x$ approaches $c\in D\cup \{-\infty ,+\infty \}$ is $L\in \mathbb{R}$ and $f(x)\le h(x)\le g(x)$ for all $x\in D$, then $h$ also approaches $L$ for $x\to c$.

${\lim }_{c\to x}f(x)={\lim }_{c\to x}h(x)={\lim }_{x\to c}g(x)=L$

PROOF

TODO