Applications of limits

Limits at infinity and one-sided limits are not merely abstract exercises — they routinely answer a natural real-world question: "what is the extreme (maximum or minimum, long-run or boundary) behaviour of this quantity?"

Maximum processing rate (Example 9.25). If the rate $r$ at which the liver removes alcohol from the bloodstream, as a function of the blood alcohol concentration $x$, follows $r(x)=\dfrac{\alpha x}{\beta+x}$ for positive constants $\alpha,\beta$, then since $r$ increases with $x$, the maximum possible rate is obtained by letting concentration grow without bound:

(dividing numerator and denominator by $x$ — the technique of Section 9.2.5). So the liver's processing rate can never exceed $\alpha$, however high the concentration climbs — a genuine physical ceiling read directly off a limit at infinity.

A one-sided limit forced by physics (Example 9.26). Einstein's relativistic mass formula $m=\dfrac{m_0}{\sqrt{1-v^2/c^2}}$ (where $c$ is the speed of light) is only physically meaningful for $v<c$, so asking what happens "as $v$ approaches $c$" only makes sense as a left-hand limit, $v\to c^-$; a genuine two-sided limit is not even defined, since $v>c$ would make the quantity under the square root negative. As $v\to c^-$, the denominator $\sqrt{1-v^2/c^2}\to0^+$ (through positive values only), so $m\to\infty$: mass grows without bound as an object's speed nears the speed of light — exactly the physical statement that no massive object can actually be pushed to reach $c$.

Terminal velocity (Example 9.27). A falling object's velocity model $r(t)=\dfrac{32}{k}\cdot\dfrac{1-e^{-32kt}}{1+e^{-32kt}}$ involves an exponential term $e^{-32kt}$ that vanishes as $t\to\infty$ (for $k>0$), so the long-run ("terminal") velocity is found simply by letting that exponential term tend to $0$ inside the formula:

the speed at which air resistance and gravity come into balance. …