Question

What are some examples of end behavior?

Answer 1

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(x)=c` for every `x`, then of course also the limit as `x` approaches `\pm\infty` will still be `c`.

Polynomials

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

• Even degree: polynomials of even degree tend to `+\infty` no matter which direction `x` is approaching to, so you have that
  `lim_{x\to\pm\infty} f(x)=+\infty`, if `f(x)` 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\to-\infty} a^x = +\infty`
`lim_{x\to\infty} a^x = 0`

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

`lim_{x\to-\infty} a^x = 0`
`lim_{x\to\infty} a^x = +\infty`

Logarithms

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

`lim_{x\to+\infty} log_a(x)=0`

while if `a>1`

`lim_{x\to+\infty} log_a(x)=+\infty`

Roots

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