Applications of Limits
We often want to study the long-run behaviour of a physical process for which a mathematical model has been devised.
As the parameter in the model increases, as time or the variable controlling the process becomes large, we investigate whether the model approaches a limiting state. That is, we look for asymptotic behaviour.
Example 1
Investigate the behaviour of $f(x)=\frac{3x}{1+x^2}$f(x)=3x1+x2 as $x$x increases.
As we are not at the moment concerned with the value $x=0$x=0, we can safely divide the numerator and the denominator by $x^2$x2 in order to make use of a known limit fact. Thus, $f(x)=\frac{\frac{3}{x}}{\frac{1}{x^2}+1}$f(x)=3x1x2+1. Clearly, $f(x)\rightarrow0$f(x)→0 as $x\rightarrow\pm\infty$x→±∞, because $\lim_{x\rightarrow\infty}\frac{1}{x}=0$limx→∞1x=0.
We can learn even more without drawing the function. If $x>0$x>0, $f(x)=\frac{3x}{1+x^2}$f(x)=3x1+x2 is positive, while if $x<0$x<0, $f(x)$f(x) is negative. We also have $f(0)=0$f(0)=0.
The denominator is never zero, so there cannot be a vertical asymptote. Moreover, the function is an odd function, since $f(-x)=-f(x)$f(−x)=−f(x) for all $x$x. This means the graph will have rotational symmetry about the origin. So, the graph of the function $f$f must have a shape something like the following.
Example 2
Investigate the asymptotic behaviour of $g(x)=\frac{x^3+x+1}{x^2+1}$g(x)=x3+x+1x2+1.
It seems apparent that since the numerator is a polynomial of higher degree than that in the denominator, the function has no upper or lower bound as $x$x increases. By the division algorithm or by noticing that $g(x)=\frac{x(x^2+1)+1}{x^2+1}$g(x)=x(x2+1)+1x2+1, we can rewrite the function as $g(x)=x+\frac{1}{x^2+1}$g(x)=x+1x2+1.
We see that the fraction part can be made vanishingly small by allowing $x$x to be large enough in the positive or negative direction. Thus, $g(x)\rightarrow x$g(x)→x as $x\rightarrow\pm\infty$x→±∞.
The line $h(x)=x$h(x)=x is an asymptote for $g(x)$g(x). It has a gradient of $1$1.
Example 3
Expressions involving exponentials can lead to functions that exhibit asymptotic behaviour. A simple example is the function defined on the real numbers, given by $f(x)=e^{-x}$f(x)=e−x. Since this is just $\frac{1}{e^x}$1ex and $e^x\rightarrow\infty$ex→∞ as $x\rightarrow\infty$x→∞, it follows that $f(x)\rightarrow0$f(x)→0 as $x\rightarrow\infty$x→∞. That is, the function has a horizontal asymptote.
We might investigate more complicated expressions involving $e^x$ex, such as $g(x)=\frac{e^x}{\pi-e^x}$g(x)=exπ−ex.
We note that for very large $x$x, the numerator and denominator are almost the same, except for sign. We can rewrite the expression as $g(x)=\frac{1}{\frac{\pi}{e^x}-1}$g(x)=1πex−1. Then, making use of the fact that $\frac{1}{y}\rightarrow0$1y→0 as $y\rightarrow\infty$y→∞, we see that $g(x)\rightarrow\frac{1}{0-1}=-1$g(x)→10−1=−1 as $x\rightarrow\infty$x→∞.
Note that $g(0)=\frac{1}{\pi-1}$g(0)=1π−1 and when $x=\ln\pi$x=lnπ the function is undefined. Moreover, the closer $x$x gets to $\ln\pi$lnπ, the larger the function value becomes - values just below $\ln\pi$lnπ give very large, negative function values, and values just above give very large, positive function values.
When $x\rightarrow-\infty$x→−∞, $g(x)\rightarrow0$g(x)→0 because $e^x\rightarrow0$ex→0 and $\frac{\pi}{e^x}\rightarrow\infty$πex→∞.
Thus, $g(x)$g(x) has horizontal asymptotes at $0$0 and $-1$−1 and a vertical asymptote at $x=\ln\pi$x=lnπ.
Example 4
Find the asymptotes of $f\left(x\right)=\frac{e^x}{e^3-e^x}$f(x)=exe3−ex.
Write the equations of all asymptotes on the same line, separated by commas.
Example 5
The population $P$P of stray cats in a town can be modelled by $P\left(t\right)=\frac{1}{\left(0.997-\frac{t}{29}\right)^{29}}$P(t)=1(0.997−t29)29, where $t$t is in months.
The $t$t-value of the vertical asymptote of the function $P(t)$P(t) is called the 'doomsday' value, since the number of stray cats grows infinitely large when $t$t approaches this value.
Find the doomsday value for this town.
Further research showed that this model is appropriate only for the first $24$24 months, and after this point the population growth will slow and begin to taper off. Which of the following graphs best represents a model which incorporates this new information? Choose the most appropriate answer.
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D
Example 6
Which function has the properties $\lim_{x\to-4^+}f\left(x\right)=-\infty$limx→−4+f(x)=−∞ and $\lim_{x\to-4^-}f\left(x\right)=-\infty$limx→−4−f(x)=−∞?
$f\left(x\right)=\frac{1}{4-x}$f(x)=14−x
A$f\left(x\right)=\frac{x}{\left(x+4\right)^2}$f(x)=x(x+4)2
B$f\left(x\right)=\frac{1}{\left(x+4\right)^2}$f(x)=1(x+4)2
C$f\left(x\right)=\frac{1}{x+4}$f(x)=1x+4
D