The Limit

At the boards in How Fast Right Now?, your group computed average speeds over shrinking intervals and watched the answers march: $7$, $6.1$, $6.01$, $6.001$ — each one closer to something, none of them ever arriving. Then someone in your group said the sentence every group eventually says: “it’s obviously going to 6.” The limit is mathematics agreeing with that sentence, and giving it a notation:

${\lim }_{h\to 0}\frac{f(3+h)-f(3)}{h}=6$

Naming the destination

A limit is the value a quantity approaches, whether or not it ever gets there. That last clause is the whole idea. No secant your group drew had slope exactly 6 — every one of them needed two separate points, and two separate points always leave a gap $h$ between them. The limit is not the best secant. It is the destination the secants agree on, and it is exactly where Zooming In was pointing when the curve started looking like a straight line under magnification.

Self-check: no secant has slope 6 — so where does the

6 come from? (click to expand) From the pattern, not from any single measurement. Each secant slope is $6+h$ for its own small $h$, and the numbers $6.1$, $6.01$, $6.001$ leave no doubt about where that pattern is headed as $h$ shrinks. The limit is the one value the slopes crowd around — every smaller $h$ lands you closer to 6, and nothing else has that property.

Two roads to the same limit

Your group found this limit twice, and the two routes matter.

The first road is numeric: substitute $h=1$, $h=0.1$, $h=0.01$ into $\frac{(3+h{)}^2-9}{h}$ and watch the outputs settle. Honest, slow, and convincing.

The second road is algebraic: simplify first.

$\frac{(3+h{)}^2-9}{h}=\frac{6h+h^2}{h}=6+h$

Now the destination is visible without a single substitution — as $h\to 0$, the expression $6+h$ heads straight for 6. The algebra did not change the answer; it changed how clearly you could see it. When a limit resists you later in the course, this is the move: simplify until the destination shows itself.

Limits in the wild

Limits were hiding in your mathematical life before this week. The sequence ${\left(1+\frac{1}{n}\right)}^n$ creeps toward a number near $2.718$ that will matter enormously in a few weeks. The ratio of each Fibonacci number to the one before it settles toward the golden ratio, about $1.618$. A graph sliding along an asymptote is a limit drawn in ink. None of these processes finishes — and the limit is how we speak precisely about where each one is going anyway.

The Derivative builds this idea into a definition, and Limits Practice makes the two roads a reflex. Take the numeric road first, every time — the estimate audits the algebra.

Curriculum connection

A1.3

make connections, with or without graphing technology, between an approximate value of the instantaneous rate of change at a given point on the graph of a smooth function and average rates of change over intervals containing the point (i.e., by using secants through the given point on a smooth curve to approach the tangent at that point, and determining the slopes of the approaching secants to approximate the slope of the tangent)

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A1.4

recognize, through investigation with or without technology, graphical and numerical examples of limits, and explain the reasoning involved (e.g., the value of a function approaching an asymptote, the value of the ratio of successive terms in the Fibonacci sequence) Sample problem: Use appropriate technology to investigate the limiting value of the terms in the sequence ${\left(1+\frac{1}{1}\right)}^1$, ${\left(1+\frac{1}{2}\right)}^2$, ${\left(1+\frac{1}{3}\right)}^3$, ${\left(1+\frac{1}{4}\right)}^4$, …, and the limiting value of the series $4\times 1-4\times \frac{1}{3}+4\times \frac{1}{5}-4\times \frac{1}{7}+4\times \frac{1}{9}-\dots$.

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A1.5

make connections, for a function that is smooth over the interval $a\le x\le a+h$, between the average rate of change of the function over this interval and the value of the expression $\frac{f(a+h)-f(a)}{h}$, and between the instantaneous rate of change of the function at $x=a$ and the value of the limit ${\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$ Sample problem: What does the limit ${\lim }_{h\to 0}\frac{f(4+h)-f(4)}{h}=8$ indicate about the graph of the function $f(x)=x^2$? The graph of a general function $y=f(x)$?

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A1.6

compare, through investigation, the calculation of instantaneous rates of change at a point $(a,f(a))$ for polynomial functions [e.g., $f(x)=x^2$, $f(x)=x^3$], with and without simplifying the expression $\frac{f(a+h)-f(a)}{h}$ before substituting values of $h$ that approach zero [e.g., for $f(x)=x^2$ at $x=3$, by determining $\frac{f(3+1)-f(3)}{1}=7$, $\frac{f(3+0.1)-f(3)}{0.1}=6.1$, $\frac{f(3+0.01)-f(3)}{0.01}=6.01$, and $\frac{f(3+0.001)-f(3)}{0.001}=6.001$, and by first simplifying $\frac{f(3+h)-f(3)}{h}$ as $\frac{(3+h{)}^2-3^2}{h}=6+h$ and then substituting the same values of $h$ to give the same results]

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