Chapter 12 · Introduction

Chapter 12 — Limits and Derivatives

How fast is a ball moving at one instant, and how steep is a curve at one point? Let an interval shrink to nothing without ever dividing by zero — that is a limit — and the answer is the derivative.

A fast bowler releases the ball and the big screen flashes "$148\ \text{km/h}$". But speed is distance divided by time, and at a single instant the ball covers no distance in no time — $\tfrac00$, which means nothing. What the speed gun really does is measure how far the ball travels in a very short time, a few thousandths of a second, and divide. The shorter that time, the closer the answer comes to "the speed at that instant". Mathematics turns this idea into a precise tool: we never put the time equal to zero; instead we watch what the average speed approaches as the time interval shrinks. The number it approaches is called a limit.

The question is old. In the twelfth century the Indian astronomer Bhāskara II wrote about the tātkālika gati, the "instantaneous motion" of a planet, and astronomers of the Kerala school later found series that behave exactly like the modern derivatives of sine and cosine. In seventeenth-century Europe, Newton and Leibniz turned the idea into calculus. In this chapter you will meet its two central ideas. First, the limit $\displaystyle\lim_{x\to a}f(x)$: what $f(x)$ approaches as $x$ gets close to $a$, from the left and from the right — even when $f(a)$ itself is undefined. You will learn the algebra of limits, the quick ways to find limits of polynomials and rational functions, and two famous limits, $\displaystyle\lim_{x\to0}\tfrac{\sin x}{x}=1$ and $\displaystyle\lim_{x\to0}\tfrac{1-\cos x}{x}=0$. Second, the derivative $f'(x)$: the limit of the average rate of change $\tfrac{f(x+h)-f(x)}{h}$ as $h\to0$ — the instantaneous speed, and the slope of the tangent to a curve. You will find derivatives from first principles and then with the sum, product and quotient rules, which make light work of polynomials and trigonometric functions.

Average rate over an interval→Limit: let the interval shrink→Standard limits: $\tfrac{x^n-a^n}{x-a}$, $\tfrac{\sin x}{x}$→Derivative from first principles→Sum, product and quotient rules

🎯 By the end of this chapter you can…

explain the limit $\displaystyle\lim_{x\to a}f(x)=l$ in words, estimate it from a table or a graph, and find left-hand and right-hand limits — deciding whether a limit exists at a hole, a jump or a corner;
use the algebra of limits and find limits of polynomials and rational functions by substitution, factorising, rationalising and the result $\displaystyle\lim_{x\to a}\tfrac{x^n-a^n}{x-a}=na^{n-1}$;
prove $\displaystyle\lim_{x\to0}\tfrac{\sin x}{x}=1$ with the sandwich theorem and use it, and $\displaystyle\lim_{x\to0}\tfrac{1-\cos x}{x}=0$, to evaluate trigonometric limits;
interpret the derivative $f'(a)$ as an instantaneous rate of change and as the slope of the tangent, and find derivatives of $x^2$, $\tfrac1x$, $\sqrt x$, $\sin x$, $\cos x$ and similar functions from first principles;
differentiate polynomials and trigonometric functions quickly with the derivative of $x^n$ and the sum, difference, product and quotient rules, and use derivatives to find speeds and tangent lines.

The window is centred on a point $P$ of the curve. Press Play (or drag the zoom slider) to magnify the picture around $P$. The red chord joins $P$ to a second point of the curve near the right edge of the window. Watch the curve straighten out and the chord's slope settle on one number — the slope of the curve at $P$. That number is the derivative you will learn to compute in this chapter. ("Bend" is the largest gap between the curve and the blue line, as a share of the window height.)

curvezoom1×

window half-width $h$—

chord slope—

slope at $P$—

bend in the window—

Sections in this chapter

12.1

Intuitive Idea of Limits

From average speed to instantaneous speed; the limit of a function as $x\to a$; left-hand and right-hand limits at holes, jumps and corners; and the algebra of limits.

$\displaystyle\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=l$

Open section →12.2

Limits of Polynomial, Rational and Trigonometric Functions

Substitution, factorising and rationalising; the standard limit $\tfrac{x^n-a^n}{x-a}\to na^{n-1}$; the sandwich theorem, $\tfrac{\sin x}{x}\to1$ and $\tfrac{1-\cos x}{x}\to0$.

$\displaystyle\lim_{x\to0}\dfrac{\sin x}{x}=1$

Open section →12.3

The Derivative from First Principles

The derivative at a point as a rate of change and as the slope of a tangent; the derivative function; first-principles derivatives of $x^2$, $\tfrac1x$, $\sqrt x$, $\sin x$ and $\cos x$.

$f'(x)=\displaystyle\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$

Open section →12.4

Algebra of Derivatives

The derivative of $x^n$; the sum, difference, product and quotient rules; derivatives of polynomials and of $\tan x$, $\sec x$, $\operatorname{cosec}x$, $\cot x$.

$(uv)'=u'v+uv'$

Open section →

✅ Before you start

You need functions and their graphs from Section 2.3 — especially the modulus function $|x|$, the greatest integer function $[x]$ and functions defined in pieces — and the algebra of real functions ($f\pm g$, $fg$, $\tfrac fg$) from Section 2.4. Trigonometric limits use radian measure (Section 3.1), the unit circle and the graphs of $\sin x$ and $\cos x$ (Section 3.2), the sum formulae (Section 3.3) and the identities $1-\cos x=2\sin^2\tfrac x2$ and $\sin A-\sin B=2\cos\tfrac{A+B}2\sin\tfrac{A-B}2$ (Section 3.4). The binomial theorem of Section 7.1 gives the derivative of $x^n$, and the slope of a line and its point-slope equation (Section 9.1, Section 9.2) give the tangent line. From Class 10 Mathematics (Chapter 2, Polynomials) you need to factorise quadratics and cubics and to divide a polynomial by $x-a$; rationalising a denominator such as $\tfrac{1}{\sqrt x+\sqrt a}$ is Class 9 algebra.

← 11.2 Distance between Two Points 12.1 Intuitive Idea of Limits →