Math Insight

Limits of exponential functions at infinity

It is important to appreciate the behavior of exponential functions as the input to them becomes a large positive number, or a large negative number. This behavior is different from the behavior of polynomials or rational functions, which behave similarly for large inputs regardless of whether the input is large positive or large negative. By contrast, for exponential functions, the behavior is radically different for large positive or large negative.

As a reminder and an explanation, let's remember that exponential notation started out simply as an abbreviation: for positive integer $n$, $$2^n = 2 \times 2 \times 2\times \ldots \times 2\;\;\;\;\;\hbox{ ($n$ factors) }$$ $$10^n = 10 \times 10 \times 10\times \ldots \times 10\;\;\;\;\;\hbox{ ($n$ factors) }$$ $$\left({1\over 2}\right)^n = \left({1\over 2}\right) \times \left({1\over 2}\right) \times \left({1\over 2}\right)\times \ldots \times \left({1\over 2}\right) \;\;\;\;\;\hbox{ ($n$ factors) }$$

From this idea it's not hard to understand the fundamental properties of exponents (they're not laws at all): $$a^{m+n} = \underbrace{a\times a\times a\times \ldots \times a}_{m+n}\;\;\;\;\;\hbox{ ($m+n$ factors)}$$ $$ = \underbrace{(a\times a\times a\times \ldots \times a)}_{m} \times \underbrace{(a\times a\times a\times \ldots \times a)}_{n} = a^m \times a^n$$ and also $$a^{mn} = \underbrace{(a\times a\times a\times \ldots \times a)}_{mn} = $$ $$ = \underbrace{ \underbrace{(a\times a\times a\times \ldots \times a)}_{m} \times \ldots \times \underbrace{(a\times a\times a\times \ldots \times a)}_{m} }_n = (a^m)^n$$ at least for positive integers $m,n$. Even though we can only easily see that these properties are true when the exponents are positive integers, the extended notation is guaranteed (by its meaning, not by law) to follow the same rules.

Use of other numbers in the exponent is something that came later, and is also just an abbreviation, which happily was arranged to match the more intuitive simpler version. For example, $$a^{-1} = {1\over a}$$ and (as consequences) $$a^{-n} = a^{n\times(-1)} = (a^n)^{-1} = {1\over a^n}$$ (whether $n$ is positive or not). Just to check one example of consistency with the properties above, notice that $$a = a^1 = a^{(-1)\times (-1)} = {1\over a^{-1}} = {1\over 1/a} = a$$ This is not supposed to be surprising, but rather reassuring that we won't reach false conclusions by such manipulations.

Also, fractional exponents fit into this scheme. For example $$a^{1/2} = \sqrt{a}\;\;\;\;\;a^{1/3} = \sqrt[3]{a}$$ $$a^{1/4} = \sqrt[4]{a}\;\;\;\;\;a^{1/5} = \sqrt[5]{a}$$ This is consistent with earlier notation: the fundamental property of the $n^{\rm th}$ root of a number is that its $n^{\rm th}$ power is the original number. We can check: $$a = a^1 = (a^{1/n})^n = a$$ Again, this is not supposed to be a surprise, but rather a consistency check.

Then for arbitrary rational exponents $m/n$ we can maintain the same properties: first, the definition is just $$a^{m/n} = (\sqrt[n]{a})^m$$

One hazard is that, if we want to have only real numbers (as opposed to complex numbers) come up, then we should not try to take square roots, $4^{\rm th}$ roots, $6^{\rm th}$ roots, or any even order root of negative numbers.

For general real exponents $x$ we likewise should not try to understand $a^x$ except for $a>0$ or we'll have to use complex numbers (which wouldn't be so terrible). But the value of $a^x$ can only be defined as a limit: let $r_1,r_2,\ldots$ be a sequence of rational numbers approaching $x$, and define $$a^x = \lim_i \;a^{r_i}$$ We would have to check that this definition does not accidentally depend upon the sequence approaching $x$ (it doesn't), and that the same properties still work (they do).

The number $e$ is not something that would come up in really elementary mathematics, because its reason for existence is not really elementary. Anyway, it's approximately $$e = 2.71828182845905$$ but if this ever really mattered you'd have a calculator at your side, hopefully.

With the definitions in mind it is easier to make sense of questions about limits of exponential functions. The two companion issues are to evaluate $$\lim_{x\rightarrow +\infty}\;a^x$$ $$\lim_{x\rightarrow -\infty}\;a^x$$ Since we are allowing the exponent $x$ to be real, we'd better demand that $a$ be a positive real number (if we want to avoid complex numbers, anyway). Then $$\lim_{x\rightarrow +\infty}\;a^x = \left\{\matrix{ +\infty& \hbox{ if } & a>1 \cr 1& \hbox{ if } & a=1 \cr 0& \hbox{ if } & 0< a<1 }\right.$$ $$\lim_{x\rightarrow -\infty}\;a^x = \left\{\matrix{ 0& \hbox{ if } & a>1 \cr 1& \hbox{ if } & a=1 \cr +\infty& \hbox{ if } & 0< a<1 }\right.$$

To remember which is which, it is sufficient to use $2$ for $a>1$ and ${1\over 2}$ for $0< a<1$, and just let $x$ run through positive integers as it goes to $+\infty$. Likewise, it is sufficient to use $2$ for $a>1$ and ${1\over 2}$ for $0< a<1$, and just let $x$ run through negative integers as it goes to $-\infty$.