Part 5 · Chapter 22

Functions, Limits & Continuity

The bedrock of calculus — what a function is, what it means to approach a value, and what it takes for a curve to be drawn without lifting the pen

Fundamentals of Mathematics Prof. Mithun Mondal Reading time ≈ 42 min
i What you'll learn
  • What a function is — its domain, codomain and range — and how to find them.
  • The main types of functions: one-one, onto, even, odd, periodic, composite and inverse.
  • The meaning of a limit, with left- and right-hand limits and the test for existence.
  • The algebra of limits and the standard limits that resolve most problems.
  • Indeterminate forms and the techniques — factorise, rationalise, substitute — that crack them.
  • Continuity at a point, the kinds of discontinuity, and the Intermediate Value Theorem.
Section 22-1

What Is a Function?

A function \(f:X\to Y\) is a rule that assigns to each element of a set \(X\) exactly one element of a set \(Y\). The set \(X\) is the domain, \(Y\) the codomain, and the set of values actually attained, \(\{f(x):x\in X\}\), the range.

X Y f
Each input maps to exactly one output (the range need not fill \(Y\))
The vertical line test. A curve in the plane is the graph of a function of \(x\) exactly when no vertical line meets it more than once — one input, one output. To find a real function's domain, exclude inputs that would force division by zero, a negative under an even root, or a non-positive argument of a logarithm.
Section 22-2

Types of Functions

Functions are classified by how they pair inputs with outputs, and by their symmetry. These labels recur constantly when we invert, compose or integrate later.

① One-one & onto

One-one (injective): distinct inputs give distinct outputs. Onto (surjective): the range fills the codomain. Both together: bijective — and invertible.

② Even & odd

Even: \(f(-x)=f(x)\) (symmetric about the \(y\)-axis). Odd: \(f(-x)=-f(x)\) (symmetric about the origin).

③ Periodic

Repeats every \(T\): \(f(x+T)=f(x)\). For instance \(\sin x\) has period \(2\pi\).

④ Composite & inverse

Composite \((f\circ g)(x)=f(g(x))\). The inverse \(f^{-1}\) exists for a bijection and undoes \(f\).

▸ The pattern

With \(f(x)=2x+3\) and \(g(x)=x^2\), the composite \((f\circ g)(x)=2x^2+3\) — feed \(g\)'s output into \(f\). Order matters: \((g\circ f)(x)=(2x+3)^2\) is different.

Section 22-3

The Idea of a Limit

The limit of \(f(x)\) as \(x\to a\) is the value \(f(x)\) approaches as \(x\) gets close to \(a\) — regardless of what happens at \(a\) itself. Approaching from below gives the left-hand limit; from above, the right-hand limit.

a L LHL→ ←RHL
The limit exists when both one-sided limits agree
🎯
Existence of a limit
\(\lim_{x\to a}f(x)=L \iff \lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\)

The limit exists only when the left- and right-hand limits are equal. The value \(f(a)\) is irrelevant to the limit — the function need not even be defined at \(a\).

Section 22-4

The Algebra of Limits

Provided the individual limits exist, limits distribute over sums, products and quotients. This lets us break a complicated limit into manageable pieces.

➗
Limit laws
\(\lim(f\pm g)=\lim f\pm\lim g,\quad \lim(fg)=\lim f\cdot\lim g,\quad \lim\dfrac{f}{g}=\dfrac{\lim f}{\lim g}\)

The quotient rule requires \(\lim g\neq0\). When direct substitution gives a finite, defined value, the limit simply equals that value — the interesting cases are exactly those where it does not.

Section 22-5

Standard Limits

A short list of limits, proved once, resolves the overwhelming majority of problems. The most famous is the trigonometric limit that underlies the derivative of \(\sin x\).

1 0
\(\dfrac{\sin x}{x}\to1\) as \(x\to0\)
🧰
The standard limits toolbox
\(\lim_{x\to0}\dfrac{\sin x}{x}=1,\quad \lim_{x\to0}\dfrac{1-\cos x}{x^2}=\tfrac12,\quad \lim_{x\to0}\dfrac{e^x-1}{x}=1\)

Also essential: \(\lim_{x\to0}\dfrac{a^x-1}{x}=\ln a\), \(\lim_{x\to0}\dfrac{\ln(1+x)}{x}=1\), \(\lim_{x\to0}(1+x)^{1/x}=e\), and \(\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=na^{n-1}\).

Section 22-6

Indeterminate Forms & Techniques

When direct substitution yields an indeterminate form such as \(\tfrac00\) or \(\tfrac{\infty}{\infty}\), the expression must be reshaped before the limit reveals itself. Three algebraic moves handle most cases.

① Factor & cancel

For \(\tfrac00\) from a common root, factor numerator and denominator and cancel the vanishing factor.

② Rationalise

When square roots cause the \(\tfrac00\), multiply by the conjugate to clear them.

③ Substitute or use a standard limit

Reshape into a known standard limit, or substitute \(x=a+h\) with \(h\to0\).

Limits at infinity. For a rational function as \(x\to\infty\), divide top and bottom by the highest power of \(x\): the limit is the ratio of leading coefficients when the degrees match, \(0\) when the bottom wins, and \(\pm\infty\) when the top wins. The full list of indeterminate forms is \(\tfrac00,\ \tfrac{\infty}{\infty},\ \infty-\infty,\ 0\cdot\infty,\ 1^{\infty},\ 0^0,\ \infty^0\).
Section 22-7

Continuity at a Point

A function is continuous at \(a\) if its graph has no break there — the limit exists, the value exists, and they agree. This single three-part condition is the whole definition.

🖋️
Continuity at \(x=a\)
\(\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)\)

Equivalently, \(\lim_{x\to a}f(x)=f(a)\): the limit must exist and equal the function's value. Sums, products, quotients (where the denominator is non-zero) and composites of continuous functions are continuous, so polynomials, \(\sin x\), \(e^x\) and the like are continuous everywhere they are defined.

Section 22-8

Kinds of Discontinuity & the IVT

A function fails to be continuous at \(a\) when the three-part condition breaks. The way it breaks sorts the discontinuity into three kinds.

Removable — a hole
Jump — a step
Infinite — an asymptote

A removable discontinuity has a limit but a wrong or missing value; a jump has unequal one-sided limits; an infinite discontinuity has a one-sided limit running off to \(\pm\infty\). On a closed interval, continuity guarantees a remarkable conclusion.

k a f(b) c
IVT: a continuous \(f\) takes every value \(k\) between \(f(a)\) and \(f(b)\)
🌉
Intermediate Value Theorem
\(f\text{ continuous on }[a,b],\ k\text{ between }f(a),f(b)\ \Rightarrow\ \exists\,c\in(a,b):f(c)=k\)

A continuous curve cannot get from one height to another without passing through every height in between. In particular, if \(f(a)\) and \(f(b)\) have opposite signs, then \(f\) has a root in \((a,b)\).

Worked Examples

Putting It to Work

1 Domain and range

Problem. Find the domain and range of \(f(x)=\sqrt{4-x^2}\).

Solution. The root needs \(4-x^2\ge0\), i.e. \(-2\le x\le2\); then \(4-x^2\) ranges over \([0,4]\), so its square root ranges over \([0,2]\):

Working
\[ \text{domain}=[-2,2],\qquad \text{range}=[0,2] \]
2 Composite functions

Problem. If \(f(x)=2x+3\) and \(g(x)=x^2\), find \(f\circ g\) and \(g\circ f\).

Solution. Substitute one function into the other:

Working
\[ (f\circ g)(x)=2x^2+3,\qquad (g\circ f)(x)=(2x+3)^2=4x^2+12x+9 \]
3 A limit by factorisation

Problem. Evaluate \(\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}\).

Solution. Direct substitution gives \(\tfrac00\); factor and cancel:

Working
\[ \lim_{x\to2}\frac{(x-2)(x+2)}{x-2}=\lim_{x\to2}(x+2)=4 \]
4 A limit by rationalisation

Problem. Evaluate \(\displaystyle\lim_{x\to0}\frac{\sqrt{1+x}-1}{x}\).

Solution. Multiply by the conjugate \(\sqrt{1+x}+1\):

Working
\[ \lim_{x\to0}\frac{(1+x)-1}{x\left(\sqrt{1+x}+1\right)}=\lim_{x\to0}\frac{1}{\sqrt{1+x}+1}=\frac12 \]
5 A standard trigonometric limit

Problem. Evaluate \(\displaystyle\lim_{x\to0}\frac{\sin 5x}{\sin 3x}\).

Solution. Write each factor in \(\tfrac{\sin\theta}{\theta}\) form, which tends to \(1\):

Working
\[ \lim_{x\to0}\frac{\frac{\sin5x}{5x}\cdot5x}{\frac{\sin3x}{3x}\cdot3x}=\frac{1\cdot5}{1\cdot3}=\frac53 \]
6 Making a function continuous

Problem. Find \(k\) so that \(f(x)=\dfrac{1-\cos x}{x^2}\) for \(x\neq0\) and \(f(0)=k\) is continuous at \(0\).

Solution. Continuity needs \(k=\lim_{x\to0}f(x)\); using the standard limit:

Working
\[ k=\lim_{x\to0}\frac{1-\cos x}{x^2}=\frac12 \]
Review

Chapter Summary

✓ Functions

One input, one output; know the domain, codomain and range.

✓ Types

One-one, onto, bijective, even/odd, periodic, composite, inverse.

✓ Limits exist

When \(\lim_{x\to a^-}f=\lim_{x\to a^+}f\); value at \(a\) is irrelevant.

✓ Toolbox

\(\tfrac{\sin x}{x}\to1\), \(\tfrac{e^x-1}{x}\to1\), \((1+x)^{1/x}\to e\), and factor/rationalise for \(\tfrac00\).

✓ Continuity

\(\lim_{x\to a}f(x)=f(a)\); breaks give removable, jump or infinite discontinuities.

✓ IVT

A continuous function hits every value between \(f(a)\) and \(f(b)\).

Practice

Problems

Substitute first; if you meet an indeterminate form, factor, rationalise or call on a standard limit. Difficulty rises down the list.

  1. Find the domain of \(f(x)=\dfrac{1}{x^2-4}\).
  2. Find the domain of \(f(x)=\sqrt{x-3}\).
  3. If \(f(x)=3x-1\) and \(g(x)=x^2+1\), find \((f\circ g)(2)\).
  4. Evaluate \(\displaystyle\lim_{x\to3}\frac{x^2-9}{x-3}\).
  5. Evaluate \(\displaystyle\lim_{x\to0}\frac{\sqrt{1+x}-\sqrt{1-x}}{x}\).
  6. Evaluate \(\displaystyle\lim_{x\to0}\frac{\tan 7x}{\sin 2x}\).
  7. Evaluate \(\displaystyle\lim_{x\to0}\frac{e^{3x}-1}{x}\).
  8. Evaluate \(\displaystyle\lim_{x\to\infty}\frac{3x^2+2x+1}{x^2-5}\).
  9. Evaluate \(\displaystyle\lim_{x\to0}\frac{1-\cos 2x}{x^2}\).
  10. Find \(k\) for which \(f(x)=\dfrac{x^2-1}{x-1}\ (x\neq1),\ f(1)=k\) is continuous at \(x=1\).
  11. Examine the continuity of \(f(x)=|x|\) at \(x=0\).
  12. Show that \(f(x)=[x]\) (the greatest-integer function) is discontinuous at every integer.
Tip: always try direct substitution first — most limits are simply the function's value. Only when you hit \(\tfrac00\) or \(\tfrac{\infty}{\infty}\) do you reshape: factor a polynomial, rationalise a root, or massage the expression into \(\tfrac{\sin\theta}{\theta}\) or \(\tfrac{e^\theta-1}{\theta}\). For continuity, the single check \(\text{LHL}=\text{RHL}=f(a)\) settles everything.