What are some examples of end behavior?

Dec 12, 2015

The end behaviour of the most basic functions are the following:

Constants
A constant is a function that assumes the same value for every $x$, so if $f \left(x\right) = c$ for every $x$, then of course also the limit as $x$ approaches $\setminus \pm \setminus \infty$ will still be $c$.

Polynomials

• Odd degree: polynomials of odd degree "respect" the infinity towards which $x$ is approaching. So, if $f \left(x\right)$ is an odd-degree polynomial, you have that ${\lim}_{x \setminus \to - \infty} f \left(x\right) = - \setminus \infty$ and ${\lim}_{x \setminus \to + \infty} f \left(x\right) = + \setminus \infty$;

• Even degree: polynomials of even degree tend to $+ \setminus \infty$ no matter which direction $x$ is approaching to, so you have that
${\lim}_{x \setminus \to \setminus \pm \setminus \infty} f \left(x\right) = + \setminus \infty$, if $f \left(x\right)$ is an even-degree polynomial.

Exponentials

The end behaviour of exponential functions depends of the base $a$: if $a < 1$, then ${a}^{x}$ has the following limits:
${\lim}_{x \setminus \to - \setminus \infty} {a}^{x} = + \setminus \infty$
${\lim}_{x \setminus \to \setminus \infty} {a}^{x} = 0$

While if $a > 1$, it goes the other way around:

${\lim}_{x \setminus \to - \setminus \infty} {a}^{x} = 0$
${\lim}_{x \setminus \to \setminus \infty} {a}^{x} = + \setminus \infty$

Logarithms

Logarithms exist only if the argument is strictly greater than zero, so their only end behaviour is for $x \setminus \to + \setminus \infty$. And again, if $a < 1$ we have that

${\lim}_{x \setminus \to + \setminus \infty} {\log}_{a} \left(x\right) = 0$

while if $a > 1$

${\lim}_{x \setminus \to + \setminus \infty} {\log}_{a} \left(x\right) = + \setminus \infty$

Roots

Like logarithm, roots don't accept negative numbers as input, so their only end behaviour is for $x \setminus \to + \setminus \infty$. And the limit as $x \setminus \to + \setminus \infty$ of any root of $x$ is always $+ \setminus \infty$.