It will be useful for this section to elaborate on the term indeterminate form introduced in Section 4.2. Recall that a limit is called an indeterminate form if it cannot be directly evaluated or deterined by a simple logical analysis. I can classify indeterminate forms by their type. For now, I will look at three types. The first two types are quotient limits.
If \(f(x)\) and \(g(x)\) approach \(\pm \infty\text{,}\) the limit is called an indeterminate form of type \(\frac{\infty}{\infty}\text{.}\) If instead both \(f(x)\) and \(g(x)\) approach \(0\text{,}\) then it is an indeterminate form of type \(\frac{0}{0}\text{.}\) In both cases, I want to use algebra to factor out and cancel off the pieces of the quotients which tends to \(\infty\) or \(0\) to solve the limit. (I previously cautioned about using \(\infty\) in arithmetic or dividing by zero. Both of those cautions stand. The expression \(\frac{\infty}{\infty}\) and \(\frac{0}{0}\) used for indeterminate forms are labels, not calculations.)
The third type of indeterminate form is a difference limit.
If both \(f(x)\) and \(g(x)\) approach \(\pm \infty\text{,}\) this limit is called an indeterminate form of type \(\infty -
\infty\text{.}\) For this type, I want to use common denominator, conjugates or other algegraic tricks to reduce it to a limit of type \(\frac{\infty}{\infty}\) or type \(\frac{0}{0}\text{.}\)
In all these definitions, I could replace \(x \rightarrow
a\) with the one-sided limit or \(x \rightarrow \pm
\infty\text{;}\) the indeterminate forms are classified the same way for any type of limit.
Subsection5.2.2A Novel Technique
The calculation of infinite limits is similar to the same steps as finite limits except that the first step, evaluation, is impossible. I start at the second step and look for a simple logical analysis. If such an analysis is not forthcoming, the limit is an indeterminate form and I need to use algebra and the limit rules. The limit rules apply to infinite limits as they did to finite limits.
In addition to the algebraic methods already discussed, for infinite limits there is a powerful technique called asymptotic analysis. In practice, this is the most commonly used approach to infinite limits.
Asymptotic analysis interprets limits at infinity as a measurements of the growth of functions. The functions \(f_1(x)
= x\text{,}\)\(f_2(x) = x^2\) and \(f_3(x) = e^x\) all get very large as \(x\) gets very large; they all grow. Asymptotic analysis asks which of these functions grows faster.
The limit of a ratio of function \(\frac{f(x)}{g(x)}\) is asking essentially the same question. By looking at the ratio of two functions as \(x \rightarrow \infty\text{,}\) I am implicitly asking which grows faster. If \(g\) grows faster, then the denominator should outpace the numerator, and the limit should tend to \(0\text{.}\) If \(f\) grows faster, then the numerator should outpace the denominator and the limit should tend to \(\infty\text{.}\) If \(f\) and \(g\) have roughly the same growth, then the limit should settle to some finite value larger than \(0\text{.}\) This leads to the notion of asymptotic order.
Subsection5.2.3Asymptotic Order
In asymptotic analysis, I start with a quotient limit.
If this limit is \(0\text{,}\) then \(g\) has greater asymptotic order than \(f\text{.}\) Alternatively, I can say that \(g\) grows faster than \(f\)or \(g\) dominates \(f\) as \(x \rightarrow
\infty\text{.}\)
If this limit is \(\infty\text{,}\) then \(f\) has greater asymptotic order than \(g\text{.}\) Alternatively, I cqn say that \(f\) grows faster than \(g\)or \(f\) dominates \(g\text{.}\)
If this limit is finite but non-zero, then \(f\) and \(g\) have the same asymptotic order. Alternatively, I can say that \(f\) and \(g\) grow at the same asymptotic rate and neither dominates.
With this definition, wI can evaluate many limits by just knowing which functions have greater or lesser asymptotic order.
Subsection5.2.4An Asymptotic Ranking of Functions
Since I need to know the relative asymptotic order of functions, I want to make a ranking. There are several principles that go into such a ranking. Many of these are obvious, but some require more work to establish. The proofs of these statements are not included in this course.
A constant function \(f(x) = c\) has a lower asymptotic order than any increasing function.
Any multiple of a function \(c f(x)\) has the same asymptotic order as the original function \(f(x)\text{.}\)
The logarithm \(f(x) = \ln x\) grows slower than any function \(f(x) = x^r\) for \(r > 0\text{.}\)
The function \(f(x) = x^r\) grows slower than \(g(x) =
x^s\) as long as \(0 \lt r\lt s\text{.}\) In particular, polynomials of lower degree grow slower than polynomials of higher degree.
The exponential function \(f(x) = e^x\) grows faster than \(g(x) = x^r\) for any \(r\text{.}\)
This covers most of the common types of growing funtions. I can summarize this in a list of asymptotic orders: in the following list ‘\(f\lt g\)’ means that \(f\) grows slower than \(g\text{.}\)
\begin{equation*}
c \lt \ln x \lt \ldots \lt x^{\frac{1}{3}} \lt
x^{\frac{1}{2}} \lt x \lt x^2 \lt x^3 \lt \ldots \lt e^x
\lt \ldots
\end{equation*}
You will frequently need to reference this list for many limits in the course. There are other functions at the top of this list which grow faster than \(e^x\text{,}\) but they are not frequently used.
Subsection5.2.5Asymptotic Ranking, Sums and Product
In quotient limits, the numerator or denominator may be more complicated than the simple functions in the asymptotic ranking. There are two useful rules to help us work with \(f\) and \(g\) which are combinations of pieces of various asymptotic order.
If \(f = f_1 + f_2 + f_3 + \ldots + f_n\) then the asymptotic order of \(f\) is the maximum of the asymptotic order of the \(f_i\text{.}\) This means that in a sum or difference, I only need to consider the fastest growing pieces. I can simply ignore all the rest.
I don’t have as precise a rule for products. However, I can say that the product of two growing functions has a higher asymptotic order than either piece. For example, \(xe^x\) has greater asymptotic order than either \(e^x\) or \(x\text{.}\)
I look at the asymptotic order. If \(f\) has a greater asymptotic order, the limit is \(\infty\) or \(-\infty\) (depending on the sign of the fraction as \(x\) gets large). If \(g\) has a greater asymptotic, the limit is zero.
If \(f\) and \(g\) have the same asymptotic order, the value of the limit is the ratio of the leading coefficients. The leading coefficients are the coefficients which sit in front of the term with the highest asymptotic order in the numerator or denominator.
The order of the numerator is \(x^4\text{.}\) The order of the denominator is \(x^5\text{.}\) The denominator has a higher asymptotic order, so the limit is zero.
The asymptotic order the numerator is \(x^4\) and the denominator has the same asymptotic order. Therefore, I look at the leading coefficients. The leading coefficient in the numerator is \(8\text{;}\) in the denominator, it is \(14\text{.}\) Only these terms matter, which radically simplifies the limit.
Most of this section has focused on limits at the variables goes to \(\infty\text{.}\) This is the most common place for asymptotic analysis (particular when the variable is time and this \(t
\rightarrow \infty\) in interpreted as long-term behaviour), but I can also consider limits as the variable approaches \(-\infty\text{.}\) The same principles apply: I can use exactly the same techniques of asymptotic analysis. However, I have to be a bit careful with the behaviour of functions for negative numbers.
Many functions have the same asymptotic order going either to positive or negative \(\infty\text{.}\) This includes trig functions, polynomials, and odd roots.
Some function simply aren’t defined for a domain that goes to \(-\infty\text{.}\) Asking for limits of these functions doesn’t make sense and they don’t have an asymptotic order going to \(-\infty\text{.}\) This include logarithms and even roots.
Some functions simply have a different behaviour going to \(-\infty\text{.}\) The most important example if the exponentail. \(f(x) = e^x\) is exponential growth as \(x \rightarrow \infty\text{,}\) but exponential decay as \(x
\rightarrow \infty\text{.}\) For \(f(x) = e^{-x}\text{,}\) these are reverse. Doing asymptotic analysis with these functions is really just a matter of familiarity: knowing the behaviours of various functions and how they different in the positive and negative directions.
