Roughly, the ‘end behavior’ of a function refers to what happens to its graph as you travel ...
‘Farther and farther to the right’ means you let $\,x\,$ go to infinity,
which is written as ‘$\,x\rightarrow\infty\,$’.
‘Farther and farther to the left’ means you let $\,x\,$ go to negative infinity,
which is written as ‘$\,x\rightarrow\infty\,$’.
End behavior is studied to understand what happens to the outputs from a function
(the $y$values of points on the graph)
as you move farther and farther away from the origin.
For example, you might want to know:
In general, functions can have a variety of end behaviors, as
shown in the table below.
Only righthand end behaviors are shown here.
increasing without bound as you move to the right:
as $x\rightarrow\infty\,$,
$\,y\rightarrow\infty$ read as: ‘as $\,x\,$ goes to infinity, $\,y\,$ goes to infinity’ 




decreasing without bound as you move to the right:
as $x\rightarrow\infty\,$,
$\,y\rightarrow\infty$ read as: ‘as $\,x\,$ goes to infinity, $\,y\,$ goes to negative infinity’ 




approaching a particular real number (like $\,3\,$) as you move to the right:
as $x\rightarrow\infty\,$,
$\,y\rightarrow 3$ read as: ‘as $\,x\,$ goes to infinity, $\,y\,$ approaches $\,3\,$’ 



Polynomials can't exhibit the full variety of end behavior illustrated above.
As $\,x\,$ gets large, $\,y\,$ can only do two things:
go to infinity, or go to negative infinity.
The best news is that we only need one term of the polynomial
to figure out what the end behavior is!
For example, suppose you're investigating the end behavior of
$\,P(x) = x^3  3x^2 + 5x  10\,$.
Notice that:
Why?
A renaming of $\,P(x)\,$ does the trick,
where we factor out the highest power of $\,x\,$:
$$
\cssId{s61}{P(x) = x^3  3x^2 + 5x  10}
\cssId{s62}{= x^3\left(1  \frac 3x + \frac 5{x^2}  \frac{10}{x^3}\right)}
$$
From this new name, it is clear that as $\,x\,$ gets big, all the terms go to zero except the highest power term.
Roughly, the highest power term ‘washes out’ all the other terms when $\,x\,$ is big.
Think about this.
Even when $\,x\,$ is as small as $\,x = 50\,$,
let's investigate each of the terms in $\,P(x)\,$:
All possible end behaviors of polynomials are summarized in the following table.
END BEHAVIORS OF POLYNOMIALS Consider a polynomial with degree $\,n\,$ and leading coefficient $\,a\,$. Thus, the highest power term is $ax^n$. 

degree ($n$) 
leading coefficient ($a$) 
example  end behavior  
$n$ even: $n = 2,\, 4,\, 6,\, \ldots$ 
$a > 0$ 
as in $\,y = 3x^2\,$: $a = 3\,$, $n = 2$ 


$a < 0$ 
as in $\,y = 3x^2\,$: $a = 3\,$, $n = 2$ 


$n$ odd: $n = 1,\, 3,\, 5,\, \ldots$ 
$a > 0$ 
as in $\,y = 2x^3\,$: $a = 2\,$, $n = 3$ 


$a < 0$ 
as in $\,y = 2x^3\,$: $a = 2\,$, $n = 3$ 

Let $\,P(x) = 2x^3  5x^7 + 6x  5\,$.
Determine the end behavior of $\,P\,$.
SOLUTION:
The highest order term is $\,5x^7\,$.
For large values of $\,x\,$, $\,P(x)\approx 5x^7\,$.
Think of substituting a large positive number into $\,5x^7\,$.
The sign of the output is:
$()(+)^7 = ()$
The output is a large negative number, so:
as $\,x\rightarrow\infty,\ \ P(x)\rightarrow \infty$
Think of substituting a large negative number into $\,5x^7\,$.
The sign of the output is:
$()()^7 = ()() = (+)$
The output is a large positive number, so:
as $\,x\rightarrow\infty,\ \ P(x)\rightarrow \infty$
Mathematical sentences like ‘as $\,x\rightarrow\infty\,$, $\,y\rightarrow\infty\,$’
can be made precise in a Calculus course.
For those of you who are curious right now:
$\text{as } x\rightarrow\infty,\ y\rightarrow\infty$  is equivalent to 
for every $\,N > 0\,$ there exists $\,M > 0\,$ such that whenever $\,x > M\,$, $y > N$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
