Drop an object from far above the Earth, and it falls faster and faster—for a while.
Indeed, since acceleration due to gravity is about
$\frac{9.8 \text{ meters per second}}{\text{second}}\,$,
its speed initially increases by about 9.8 m/sec every second.
But, it doesn't keep going faster and faster.
Due to frictional forces, it settles on a socalled ‘terminal velocity’.
If you model distance traveled versus time mathematically, you'll be looking at a slant asymptote.
Functions can ‘get big’ in different ways.
When they ‘get big’ by looking more and more like a ‘slanted’ line (i.e.,
not horizontal and not vertical),
then the function is said to have a slant asymptote.
Slant asymptotes are also called oblique asymptotes.
As discussed in Introduction to Asymptotes,
an asymptote is a
curve (usually a line)
Here's how this general definition is ‘specialized’ to get a
slant asymptote: that another curve gets arbitrarily close to as $\,x\,$ approaches $\,+\infty\,$ or $\,\infty\,$.
What's the key to a slant asymptote situation? as inputs get arbitrarily large (big and positive, or big and negative).
This section gives a precise discussion of slant asymptotes, 
The red line is a slant asymptote for the blue curve. As $\,x\rightarrow\infty\,$, the blue curve approaches the red line. 
Let $\,f(x) = 3x + \text{e}^{x}\,$ (the blue curve at right). 
Let $\displaystyle\,R(x) = \frac{x^2  4x  5}{x  3}\,$.
When $\,x\,$ is big, the outputs from $\,R(x)\,$ looks like $\,\frac{\text{big}}{\text{big}}\,$, which is not very useful.
As usual, we will rename the function to better understand what happens when $\,x\,$ is big:
The key is to do a long division:
$x$  $$  $1$  
$x3$  $x^2$  $$  $4x$  $$  $5$ 
$(x^2$  $$  $3x$  $)$  
$x$  $$  $5$  
$($  $x$  $+$  $3)$  
$8$ 
Thus, $$ R(x) = \frac{x^2  4x  5}{x  3} \ \ =\ \ x  1 \ \ \ + \overbrace{\left(\frac{8}{x3}\right)}^{\text{this part tends to zero}} $$
Note that as $\,x\rightarrow\pm\infty\,$, $\displaystyle\,\frac{8}{x3}\rightarrow 0\,$.
Therefore, as inputs get big, the function is looking more and more like
the line $\,y = x  1\,$.
Thus, $\,y = x  1\,$ is a slant asymptote for the function $\,R\,$.
The graph of the function $\,\color{blue}{R}\,$ (in blue), together with its slant asymptote $\,\color{red}{y = x  1}\,$ (in red), is shown at right.
When we were doing the long division of $\,x^2  4x  5\,$ by $\,x  3\,$ up above, 
In summary, we have:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
