Functions can dominate other functions, or they can approach infinity similarly. If
the two functions approach infinity similarly, then the limit of their quotient equals
a number. The graph of the quotient approaches a horizontal asymptote.
This goes for \(0\) as well.
Consider the function \(B(t) = \frac {3e^t - 5}{e^t + 1}\).
As \(t \to \infty \), the exponential pieces will dominate the constant pieces.
As \(t \to -\infty \), we have a different story. \(\lim \limits _{t \to -\infty } e^t = 0\). Therefore, in this case, the constant terms dominate.
The graph has two different horizontal asymptotes.
![[Picture]](indeterminate-29d060cfde56e5c95691b830762ae0b8.svg)
Indeterminate Forms
We have been examining functions that approach infinity (or \(0\)) similarly. When
combine similar functions into a new quotient function, this new function might have
a contant end-behavior.
The problem is that a fraction of the form \(\frac {very \, big}{very \, big}\) could equal anything.
The same thing happens with values near \(0\)
Formulas whose values seem like they are approaching \(\frac {\infty }{\infty }\) or \(\frac {0}{0}\) are called indeterminate forms, because you can’t easily determine their values.
Both the functions \(S(t) = \sqrt {2t^3 - 5}\) and \(L(t) = t\sqrt {9t+4}\) approach \(\infty \) as \(t \to \infty \).
\(R(t) = \frac {S(t)}{L(t)} = \frac {\sqrt {2t^3 - 5}}{t\sqrt {9t+4}}\) is an indeterminate form. It is approaching the form \(\frac {\infty }{\infty }\)
If we examine the quotient in terms of dominance, then we think like...
When we encounter a formula that behaves like \(\frac {\infty }{\infty }\), then we know we are looking at the
wrong formula. Calculations that start looking like \(\frac {\infty }{\infty }\) tell us to find a different
equivalent formula.
Calculus
We are introducing the term indeterminate form here in connection to dominance and horizontal asymptotes. In Calculus, this idea will be expanded significantly to include other targets besides \(\infty \).
Both the functions \(S(t) = \sin (t)\) and \(L(t) = t\) approach \(0\) as \(t \to 0\).
Do they approach \(0\) similarly?
Let’s investigate \(R(t) = \frac {\sin (t)}{t}\)
![[Picture]](indeterminate-317cd744f60a05c936d7b6993929e920.svg)
The graph is very suggestive that \(\sin (t)\) and \(t\) approach \(0\) in exactly the same way. Their quotient approaches \(1\).
We have seen an even weirder indeterminate form.
Everyone knows that \(1\) to any power equals \(1\). Everyone knows that numbers greater than \(1\) to bigger powers get bigger.
What about something like \(\left ( 1 + \frac {1}{x} \right )^x\) ?
The base is getting closer to \(1\), but it is always greater than \(1\). The power is growing larger.
We are heading to something like \(1^{\infty }\).
And we have seen this function actually approaches the value \(e \approx 2.71828\).
How far out in the domain do you need to go for the graph to be inside the interval \([2.7, 2.8]\)?
Calculus will investigate the ideas of indeterminance in detail. We are just getting started.
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More Examples of Dominance