Limits at Infinity: Exponentials and Logarithms

A growth ranking is simply a list of functions ordered by how fast they grow when $x$ gets huge, and with it you can find limits at infinity involving exponentials and logs. The ranking $\ln x \ll x^p \ll e^x$ looks cryptic, but it's just three runners: $\ln x$ (the logarithm, slowest), $x^p$ (any positive power of $x$, in the middle) and $e^x$ (the exponential, fastest), with $\ll$ meaning "loses badly to." Whenever one runner sits on top of a fraction and another sits on the bottom, the faster one decides what happens. That's the whole lesson, really.

In the last lesson, the highest power of $x$ decided everything. Exponentials and logs aren't powers of $x$, so we need to know where they fit in. Here's the ranking for large $x$:

So $\dfrac{x^{10}}{e^x} \to 0$ and $\dfrac{\ln x}{\sqrt x} \to 0$. That can be surprising, because for moderate $x$ both fractions are large: at $x = 10$, $x^{10}$ is ten billion while $e^{10}$ is only about $22{,}000$. But $e^x$ multiplies itself by $e$ every time $x$ goes up by $1$, and no power of $x$ can keep up with that forever. (L'Hôpital's rule, later on, proves these.)

Exponentials. Treat the biggest exponential the way you treated the highest power: factor it out, or divide top and bottom by it. For $x \to \infty$, $e^{2x}$ beats $e^x$, since $e^{2x} = (e^x)^2$. For $x \to -\infty$ it flips: every $e^{kx}$ with $k > 0$ fades to $0$, so the constants are the ones left standing.

Logarithms. Use the log rules to combine everything into a single log of a ratio. Find where the ratio is heading, then take the log of that. This works because $\ln$ is continuous, which you'll see properly in the next lesson: a small change in the inside means a small change in the log.

Radioactive decay, a cooling drink and a charging capacitor all approach limits like these. And slow log growth is why doubling a database barely changes how long a binary search takes.