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 so-called ‘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). |
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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:
$x$ | $-$ | $1$ | |||
$x-3$ | $x^2$ | $-$ | $4x$ | $-$ | $5$ |
$-(x^2$ | $-$ | $3x$ | $)$ | ||
$-x$ | $-$ | $5$ | |||
$-($ | $-x$ | $+$ | $3)$ | ||
$-8$ |
Thus, $$ \cssId{s54}{R(x)} \cssId{s55}{= \frac{x^2 - 4x - 5}{x - 3}} \ \ \cssId{s56}{=\ \ x - 1 \ \ \ + \overbrace{\left(\frac{-8}{x-3}\right)}^{\text{this part tends to zero}}} $$
Note that as $\,x\rightarrow\pm\infty\,,$ $\displaystyle\,\frac{-8}{x-3}\rightarrow 0\,.$
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, |
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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:
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