Question

What are some examples of end behaviour?

Answer 1

Hint: We first explain the functions for which the end behaviour changes. We take polynomials for odd and even functions. Based on the value of $ x $ the limit value also changes.

Complete step by step solution:
The end behaviour of a function describes the behaviour of the graph of the function at the "ends" of the X-axis. In other words, the end behaviour of a function describes the trend of the graph if we look to the right end of the X-axis (as $ x\to \infty $ ) and to the left end of the X-axis (as $ x\to -\infty $ ).
We now discuss the end behaviour of the most basic functions.
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 for $ x\to \pm \infty $ will still be $ c $ .
For polynomials we can break it into two parts-
For polynomials of odd degree, we are careful about the infinity towards which $ x $ is approaching. So, if $ f\left( x \right) $ is an odd-degree polynomial, you have that $ \underset{x\to \infty }{\mathop{\lim }}\,f\left( x \right)=\infty $ and $ \underset{x\to -\infty }{\mathop{\lim }}\,f\left( x \right)=-\infty $ . For polynomials of even degree, we get $ \underset{x\to \pm \infty }{\mathop{\lim }}\,f\left( x \right)=\infty $ .

Note: The leading coefficient for the polynomial was considered to be positive. If that changes to negative then the final values for the polynomials would change their signs too with respect to the values for positive coefficients.