The Timpson Lyceum · Calculus 1 (2024)

Limits, Rates of Change, and Main Ideas of Calculus - Lecture 4 (Calculus 1)

Overview

This lecture opens Chapter 2 and marks the transition from review material into calculus proper. It introduces the four central ideas of the subject — limits, continuity, derivatives, and integrals — at an intuitive level, then zooms in on the two ideas the course tackles first: limits and derivatives. The heart of the lecture is the realization that a derivative (a slope at a single point, or an instantaneous speed) is computed as the limit of an average rate of change as the interval shrinks to zero. After developing the difference-quotient machinery on both geometric (slope) and physical (free fall) examples, the lecture turns to evaluating limits directly: the limit laws, the "plug-in" rule for polynomials and rational functions, the factor/conjugate tricks for indeterminate forms, and the Squeeze Theorem.

Key Concepts

The four big ideas (intuition)

Limit: what a function gets arbitrarily close to as the input approaches a point — not necessarily the function's value there. Limits let us "smooth over" bad points (e.g. holes, near-divisions by zero).
Continuity: a function you can draw without lifting your pencil — no holes, jumps, or gaps.
Derivative: the slope of the function at a single point (the slope of the tangent line there).
Integral: the area under the curve over an interval.
Everything else is built on limits, so the limit is the most important idea.

Limit vs. value at a point

For the piecewise function $f(x) = 2^x$ for $x \ne 1$ and $f(1) = 1$ : the value is $f(1)=1$ , but $\lim_{x\to 1} f(x) = 2$ , because just to either side of $x=1$ the curve is essentially at $2^1 = 2$ .
For most familiar functions the limit equals the value; a difference only appears at "weird" points (holes, undefined points).

Lines and slope notation

A line needs two points. Using $\Delta$ for "change": $\Delta x = x_2 - x_1$ (end minus start), $\Delta y = y_2 - y_1$ .
Slope = rise over run: $\displaystyle m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$ .
Point-slope form: $y - y_1 = m(x - x_1)$ . Rearranging gives slope-intercept form $y = mx + b$ , where $b$ is the $y$ -value at $x = 0$ (the $y$ -intercept).

Slope at a point = limit of a difference quotient

A slope needs two points, so to get the slope at a single point $x_1$ we take a nearby second point, compute the slope of the secant line, and let the gap shrink.
With $\Delta x$ the gap (so $x_2 = x_1 + \Delta x$ ), the average rate of change is $\frac{f(x_1 + \Delta x) - f(x_1)}{\Delta x}.$
The slope (derivative) at $x_1$ is the limit as the gap goes to zero: $\text{slope at } x_1 = \lim_{\Delta x \to 0} \frac{f(x_1 + \Delta x) - f(x_1)}{\Delta x}.$
This looks like dividing by zero, but the limit lets us simplify before taking the limit, dodging that problem.

Average vs. instantaneous speed (free fall)

Galileo's law (ignoring air resistance, in feet): an object falls $f(t) = 16t^2$ feet after $t$ seconds. So $f(1)=16$ , $f(2)=64$ .
Average speed over a time interval = total distance / total time = $\dfrac{\Delta y}{\Delta t}$ .
Instantaneous speed at time $t$ is the limit of the average speed as the interval shrinks to zero — exactly the same limit-of-a-difference-quotient idea, with time in place of $x$ .

The author's $h$ -notation (same idea)

Let $h = \Delta t$ (or $\Delta x$ ). Then the average rate of change of $y = f(x)$ over $[x_1, x_1 + h]$ is $\frac{f(x_1 + h) - f(x_1)}{h},$ and the instantaneous rate is the limit as $h \to 0$ . This is the standard textbook notation used throughout.

Limit laws — given $\lim_{x\to c} f(x) = L$ and $\lim_{x\to c} g(x) = M$ :

Sum / difference: limit of a sum (difference) is the sum (difference) of the limits: $L \pm M$ .
Constant multiple: $\lim_{x\to c} k\,f(x) = kL$ — constants pull out of limits.
Product: $\lim_{x\to c} f(x)g(x) = LM$ .
Quotient: $\lim_{x\to c} \dfrac{f(x)}{g(x)} = \dfrac{L}{M}$ , provided $M \ne 0$ .
Power: $\lim_{x\to c} [f(x)]^n = L^n$ . Root: $\lim_{x\to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$ .
Identity & constant functions: $\lim_{x\to c} x = c$ and $\lim_{x\to c} k = k$ .

Plug-in rules for polynomials and rationals

If $P$ is a polynomial, $\lim_{x\to c} P(x) = P(c)$ — just substitute.
If $P, Q$ are polynomials and $Q(c) \ne 0$ , then $\lim_{x\to c} \dfrac{P(x)}{Q(x)} = \dfrac{P(c)}{Q(c)}$ .

Indeterminate forms ( $0/0$ ) — when substitution gives $0/0$ , don't reach for a calculator:

Factor and cancel the troublesome term (often via difference of squares or common factoring), since for a limit $x$ is near $c$ but never equal to $c$ , so you're not actually dividing by zero.
Multiply by the conjugate to clear a radical that's causing the trouble.

The Squeeze Theorem (Theorem 4)

If $g(x) \le f(x) \le h(x)$ for all $x$ in an open interval containing $c$ (except possibly at $c$ itself), and $\lim_{x\to c} g(x) = \lim_{x\to c} h(x) = L$ , then $\lim_{x\to c} f(x) = L$ .
The values at $c$ don't matter — only the behavior arbitrarily close to $c$ .
Consequences proved with it (using $-|\theta| \le \sin\theta \le |\theta|$ and $0 \le 1-\cos\theta \le |\theta|$ ): $\lim_{\theta\to 0} \sin\theta = 0, \qquad \lim_{\theta\to 0} \cos\theta = 1.$
Also: if $\lim_{x\to c} |f(x)| = 0$ then $\lim_{x\to c} f(x) = 0$ , since $-|f(x)| \le f(x) \le |f(x)|$ .

Order-preservation (final theorem)

If $f(x) \le g(x)$ near $c$ (except possibly at $c$ ) and both limits exist, then $\lim_{x\to c} f(x) \le \lim_{x\to c} g(x)$ .

Worked Example

Slope of $f(x) = x^2 - 1$ at the point $x = 1$ . Apply the slope-at-a-point formula, then expand, cancel the $\Delta x$ , and take the limit:

\begin{aligned} \text{slope} &= \lim_{\Delta x \to 0} \frac{f(1+\Delta x) - f(1)}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{\big[(1+\Delta x)^2 - 1\big] - \big[1^2 - 1\big]}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{1 + 2\,\Delta x + \Delta x^2 - 1 - 0}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{2\,\Delta x + \Delta x^2}{\Delta x} = \lim_{\Delta x \to 0} (2 + \Delta x) = 2. \end{aligned}

A $0/0$ limit cleared with a conjugate: $\displaystyle \lim_{x\to 0} \frac{\sqrt{x^2 + 100} - 10}{x^2}$ . Multiply by the conjugate to remove the radical, then cancel $x^2$ :

\begin{aligned} \lim_{x\to 0} \frac{\sqrt{x^2+100}-10}{x^2} &= \lim_{x\to 0} \frac{\sqrt{x^2+100}-10}{x^2}\cdot\frac{\sqrt{x^2+100}+10}{\sqrt{x^2+100}+10} \\ &= \lim_{x\to 0} \frac{(x^2+100) - 100}{x^2\big(\sqrt{x^2+100}+10\big)} \\ &= \lim_{x\to 0} \frac{x^2}{x^2\big(\sqrt{x^2+100}+10\big)} = \lim_{x\to 0} \frac{1}{\sqrt{x^2+100}+10} \\ &= \frac{1}{\sqrt{100}+10} = \frac{1}{20}. \end{aligned}

Summary

The four ideas of calculus are the limit, continuity, the derivative (a slope), and the integral (an area); all rest on the limit.
A limit is what a function approaches near a point, which can differ from the function's value there — that is exactly how limits "smooth over" holes and apparent divisions by zero.
A derivative is a special limit: the limit of the average rate of change $\frac{f(x_1+h)-f(x_1)}{h}$ as $h \to 0$ , giving the slope at a single point (or, with time, an instantaneous speed).
For free fall $f(t)=16t^2$ , average speed over an interval is $\Delta y/\Delta t$ , and shrinking the interval to zero yields the instantaneous speed.
Evaluate limits with the limit laws; for polynomials and (nonzero-denominator) rationals just substitute. For $0/0$ forms, factor and cancel or multiply by the conjugate to remove the offending factor first.
The Squeeze Theorem pins a function trapped between two others sharing a common limit; it yields $\lim_{\theta\to 0}\sin\theta = 0$ and $\lim_{\theta\to 0}\cos\theta = 1$ .