Mathematics · Chapter 6
Study notes aligned to the official NEB syllabus.
Calculus is the mathematics of change. It rests on a single idea, the limit, which lets us talk precisely about what a function approaches, and from that one idea grow the two great operations of the subject: differentiation, which measures the instantaneous rate of change (the slope of a curve), and integration, which reverses differentiation and measures accumulated quantities such as area under a curve. This chapter builds the tower in order: limits and continuity first, then the derivative with all its rules and applications, then the integral. Every result later in your mathematics and physics silently uses these tools, so master the definitions and practise the standard forms until they are automatic.
Let $f(x)$ be defined near $x = a$ (it need not be defined at $a$). We say the limit of $f(x)$ as $x$ approaches $a$ is $L$, written
$$\lim_{x \to a} f(x) = L,$$
if $f(x)$ can be made as close to $L$ as we please by taking $x$ sufficiently close to $a$ on either side, but not equal to $a$. The value of $f$ exactly at $a$ is irrelevant to the limit; what matters is the behaviour near $a$.
The (two-sided) limit exists if and only if both one-sided limits exist and are equal:
$$ \begin{aligned} \lim_{x \to a} f(x) &= L \iff \lim_{x \to a^-} f(x) \ &= \lim_{x \to a^+} f(x) \ &= L. \end{aligned} $$
If the two one-sided limits differ, the limit does not exist.
Worked example. For $f(x) = 3x + 5$, as $x \to 2$ the function approaches $3(2) + 5 = 11$, so $\displaystyle\lim_{x \to 2}(3x+5) = 11$. Here the function is also defined at $x=2$ with the same value, but that agreement is not required for the limit to exist.
When we try to evaluate a limit by direct substitution, we sometimes reach an expression that has no definite value, such as
$$\frac{0}{0}, \quad \frac{\infty}{\infty}, \quad \infty - \infty, \quad 0 \times \infty, \quad 1^{\infty}, \quad 0^0, \quad \infty^0.$$
These are the indeterminate forms. They do not mean the limit fails to exist; they mean the form itself carries no information and the expression must first be simplified (by factorising, rationalising, dividing by the highest power, or using a standard limit) before the limit can be read off.
Worked example ($\tfrac{0}{0}$ by factorising).
$$ \begin{aligned} \lim_{x \to 2} \frac{x^2 - 4}{x - 2} &= \lim_{x \to 2} \frac{(x-2)(x+2)}{x - 2} \ &= \lim_{x \to 2} (x + 2) = 4. \end{aligned} $$
Direct substitution gives $\tfrac{0}{0}$; cancelling the common factor $(x-2)$ removes the indeterminacy.
Worked example ($\tfrac{\infty}{\infty}$ by dividing).
$$ \begin{aligned} \lim_{x \to \infty} \frac{3x^2 + 2x}{5x^2 - 1} &= \lim_{x \to \infty} \frac{3 + \frac{2}{x}}{5 - \frac{1}{x^2}} \ &= \frac{3 + 0}{5 - 0} = \frac{3}{5}. \end{aligned} $$
Dividing numerator and denominator by the highest power $x^2$ makes the small terms vanish.
If $\displaystyle\lim_{x \to a} f(x) = L$ and $\displaystyle\lim_{x \to a} g(x) = M$ both exist, then limits behave well under the arithmetic operations:
$$ \begin{aligned} \lim_{x \to a} \big[f(x) \pm g(x)\big] &= L \pm M, \ \lim_{x \to a} \big[f(x), g(x)\big] &= L,M, \ \lim_{x \to a} \big[k,f(x)\big] &= k,L, \ \lim_{x \to a} \frac{f(x)}{g(x)} &= \frac{L}{M} \quad (M \neq 0). \end{aligned} $$
There is also a very useful standard algebraic limit, valid for every rational index $n$:
$$\lim_{x \to a} \frac{x^n - a^n}{x - a} = n,a^{n-1}.$$
Worked example. $\displaystyle\lim_{x \to 3} \frac{x^5 - 3^5}{x - 3} = 5 \cdot 3^{4} = 5 \times 81 = 405.$
Certain limits recur so often that they are quoted as standard results. The trigonometric ones require the angle to be measured in radians.
Trigonometric standard limits.
$$ \begin{aligned} \lim_{x \to 0} \sin x &= 0, \qquad \lim_{x \to 0} \cos x = 1, \ \lim_{x \to 0} \frac{\sin x}{x} &= 1, \qquad \lim_{x \to 0} \frac{\tan x}{x} = 1, \ \lim_{x \to 0} \frac{1 - \cos x}{x} &= 0, \ \lim_{x \to 0} \frac{\sin^{-1} x}{x} &= 1, \qquad \lim_{x \to 0} \frac{\tan^{-1} x}{x} = 1. \end{aligned} $$
Exponential and logarithmic standard limits.
$$ \begin{aligned} \lim_{x \to 0} e^x &= 1, \qquad \lim_{x \to 0} \frac{e^x - 1}{x} = 1, \ \lim_{x \to 0} \frac{a^x - 1}{x} &= \log_e a, \qquad \lim_{x \to 0} \frac{\log(1 + x)}{x} = 1, \ \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x} &= e, \qquad \lim_{x \to 0} (1 + x)^{1/x} = e, \ \lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^{x} &= e^{a}. \end{aligned} $$
Worked example (trigonometric).
$$ \begin{aligned} \lim_{x \to 0} \frac{\sin 5x}{3x} &= \lim_{x \to 0} \frac{\sin 5x}{5x} \cdot \frac{5}{3} \ &= 1 \cdot \frac{5}{3} = \frac{5}{3}. \end{aligned} $$
Worked example (exponential). $\displaystyle\lim_{x \to 0} \frac{e^{3x} - 1}{x} = \lim_{x \to 0} \frac{e^{3x} - 1}{3x}\cdot 3 = 1 \cdot 3 = 3.$
Calculus is the mathematics of change. It rests on a single idea, the limit, which lets us talk precisely about what a function approaches, and from that one idea grow the two great operations of the subject: differentiation, which measures the instantaneous rate of change (the slope of a curve), and integration, which reverses differentiation and measures accumulated quantities such as area under a curve. This chapter builds the tower in order: limits and continuity first, then the derivative with all its rules and applications, then the integral. Every result later in your mathematics and physics silently uses these tools, so master the definitions and practise the standard forms until they are automatic.
Let be defined near (it need not be defined at ). We say the limit of as approaches is , written
if can be made as close to as we please by taking sufficiently close to on either side, but not equal to . The value of exactly at is irrelevant to the limit; what matters is the behaviour near .
The (two-sided) limit exists if and only if both one-sided limits exist and are equal:
If the two one-sided limits differ, the limit does not exist.
Worked example. For , as the function approaches , so . Here the function is also defined at with the same value, but that agreement is not required for the limit to exist.
When we try to evaluate a limit by direct substitution, we sometimes reach an expression that has no definite value, such as
These are the indeterminate forms. They do not mean the limit fails to exist; they mean the form itself carries no information and the expression must first be simplified (by factorising, rationalising, dividing by the highest power, or using a standard limit) before the limit can be read off.
Worked example ( by factorising).
Direct substitution gives ; cancelling the common factor removes the indeterminacy.
Worked example ( by dividing).
Dividing numerator and denominator by the highest power makes the small terms vanish.
If and both exist, then limits behave well under the arithmetic operations:
There is also a very useful standard algebraic limit, valid for every rational index :
Worked example.
Certain limits recur so often that they are quoted as standard results. The trigonometric ones require the angle to be measured in radians.
Trigonometric standard limits.
Exponential and logarithmic standard limits.
Worked example (trigonometric).
Worked example (exponential).