What are some examples of end behavior?

Question

Theodore Amidon · Accepted Answer

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#.

Lily Anderson · Answer

undefined Some examples of end behavior in functions include:

1. Asymptotic behavior: A function approaches a certain value as x approaches positive or negative infinity.
2. Horizontal asymptote: The function levels off as x approaches positive or negative infinity.
3. Vertical asymptote: The function approaches positive or negative infinity as x approaches a certain value.
4. Behavior at a specific end: The function may approach a specific value or behave in a particular manner as x approaches a specific end of its domain.