Finite and infinite limits

Finite limits

We will start by showing a small summary of the properties of finite limits.

Let's suppose that $\displaystyle\lim_{x \to \pm \infty}{f(x)}=a$ and that $\displaystyle\lim_{x \to \pm \infty}{g(x)}=b$, then also:

If $\displaystyle\lim_{x \to {+} \infty}{f(x)}=3$ and $\displaystyle\lim_{x \to {+} \infty}{g(x)}=-5$ then:

  1. $\displaystyle\lim_{x \to {+} \infty}{f(x)+g(x)}=3-5=-2$
  2. $\displaystyle\lim_{x \to {+} \infty}{f(x)-g(x)}=3-(-5)=8$
  3. $\displaystyle\lim_{x \to {+} \infty}{f(x) \cdot g(x)}=3 \cdot (-5)=-15$
  4. $\displaystyle\lim_{x \to {+} \infty}{f(x)^g(x)}=3^{-5}=\frac{1}{3^5}=\frac{1}{243}$
  5. $\displaystyle\lim_{x \to {+} \infty}{g(x)^{f(x)}}$ does not exist since $g(x) < 0$ for $x$ large enough.
  6. $\displaystyle\lim_{x \to {+} \infty}{\sqrt[3]{g(x)}}=\sqrt[3]{-5}=-\sqrt[3]{5}$
  7. $\displaystyle\lim_{x \to {+} \infty}{\sqrt{g(x)}}$ does not exist since $g(x) < 0$ for $x$ large enough.

Infinite limits

Let's start by defining what an infinite limit of a function is $f(x)$:

$$\displaystyle\lim_{x \to {+} \infty}{f(x)}=+\infty \Longleftrightarrow \mbox{ given any } k, \mbox{ there exists another number } h$$

$$\mbox{ such that if } x>h \mbox{ then } f(x)>k$$

Intuitively, it means that we can have $f(x)$ as big as we want by choosing a sufficiently large $x$.

Similarly, we define:

$$\displaystyle\lim_{x \to {+} \infty}{f(x)}=-\infty \Longleftrightarrow \mbox{ given any } k, \mbox{ there exists another number } h$$

$$\mbox{ such that if } x>h \mbox{ then } f(x)<-k$$

and for limits when $x$ goes to minus infinity:

$$\displaystyle\lim_{x \to {-} \infty}{f(x)}=+\infty \Longleftrightarrow \mbox{ given any } k, \mbox{ there exists a another number } h$$

$$\mbox{ such that if } x<-h \mbox{ then } f(x)>k$$

$$\displaystyle\lim_{x \to {-} \infty}{f(x)}=-\infty \Longleftrightarrow \mbox{ given any } k, \mbox{ there exists a another number } h $$

$$\mbox{ such that if } x<-h \mbox{ then } f(x)<-k$$

Let's see three basic examples of functions that tend to infinity:

$k$th power: if $k>0, \displaystyle\lim_{x \to {+}\infty}{x^k}=+\infty$

and in particular $\displaystyle\lim_{x \to {+}\infty}{p\cdot x^k}=sign(p) \cdot \infty$ , where $p$ is a real value other zero.

From this point, we deduce that the polynomial functions tend to infinity as $x$ becomes larger.

image

In this example we can see the function $f(x)=3x^4$. When $x$ becomes large, the function grows to infinity.

Exponential: if $a>1, \displaystyle\lim_{x \to {+}\infty}{a^x}=+\infty$

and likewise if $a>1, \displaystyle\lim_{x \to {+}\infty}{p \cdot a^x}=sign(p) \cdot \infty$ .

An example for this case is the function $f(x)=\dfrac{1}{2}e^x$. It tends to infinity as $x$ tends to infinity.

Logarithmic: if $a>1, \displaystyle\lim_{x \to {+}\infty}{\log_{a}x}=+ \infty$

Similarly if $a>1 \displaystyle\lim_{x \to {+}\infty}{\log_{a}x}=sign(p) \cdot \infty$.

For example, the function $f(x)= \log_{e}x=\ln x$. This function tends to infinity as $x$ becomes very large.

Infinite's arithmetic

Let's suppose that $\displaystyle\lim_{x \to {+} \infty}{f(x)}=+\infty$ and that $\displaystyle\lim_{x \to {+} \infty}{g(x)}=+\infty$, then we observe without problems that:

$$\displaystyle\lim_{x \to {+} \infty}{f(x)+g(x)}=\displaystyle\lim_{x \to {+} \infty}{f(x)}+\displaystyle\lim_{x \to {+} \infty}{g(x)}=+\infty + \infty=+\infty$$

$$\displaystyle\lim_{x \to {+} \infty}{f(x) \cdot g(x)}=\displaystyle\lim_{x \to {+} \infty}{f(x)} \cdot \displaystyle\lim_{x \to {+} \infty}{g(x)}=(+\infty) \cdot (+\infty)=+\infty$$

However, we will have problems when we encounter situations like the following one:

$$\displaystyle\lim_{x \to {+} \infty}{f(x)-g(x)}=\displaystyle\lim_{x \to {+} \infty}{f(x)}-\displaystyle\lim_{x \to {+} \infty}{g(x)}=(+\infty)-(+\infty)$$

since to if we subtract infinity from infinity it gives us an indeterminacy.

Similarly, we might ask ourselves about these properties when we a function with an infinite limit and one with a finite limit.

Let's see a small table that will show us how to work when we have different kinds necessary to produce infinity with other infinites and with finite limits:

SUMS PRODUCTS
$(+\infty)+a=+\infty$ $(+\infty)\cdot(+\infty)=+\infty$
$(+\infty)+(+\infty)=+\infty$ $(+\infty)\cdot(-\infty)=-\infty$
$(-\infty)+a=-\infty$ $(+\infty)\cdot a=sign(a) \cdot \infty$
$(-\infty)+(-\infty)=-\infty$ $(-\infty)\cdot a=-sign(a) \cdot \infty$
$-(-\infty)=+\infty$  
DIVISIONS POWERS
$\frac{a}{\pm \infty}=0$ $(+\infty)^{+\infty}=+\infty$
$\frac{a}{0}=\pm \infty$ if $a\neq 0$ $(+\infty)^{-\infty}=0$
$\frac{\pm \infty}{0}=\pm \infty$ if $a$ > $0$ $$(+\infty)^a= + \infty$$
$\frac{0}{\pm \infty}=0$ if $a$ < $0$ $$(+\infty)^a=0$$
  if $a\neq 0$ $$a^0=1$$
  if $a$ > $1$ $$a^{+\infty}=+\infty \\ a^{-\infty}=0$$
  if $0$ <$a$ < $1$ $$a^{+\infty}=0 \\ a^{-\infty}=+ \infty$$

These operations can be realized after finding the limits of the functions involved.

Nevertheless, the operations that are not in the table can produce indeterminacies, for example, the following expressions:

$$(+\infty)-(+\infty) \\ 0 \cdot (\pm \infty) \\ \frac{0}{0} \\ (+\infty)^0 \\ 1^{\pm \infty} \\ 0^0 \\ \frac{\pm \infty}{\pm \infty}$$

Practice exercises