NDA Maths · Limits & Continuity

Evaluating Limits — Factor, Rationalise, Standard Forms

The core toolkit for a 0/0 limit: direct substitution first, then factor-and-cancel, rationalise a surd, apply a standard limit, use L'Hôpital, or handle the 1^∞ form via the exponential trick.

All Limits & Continuity notes NDA Maths strategy

Why this matters

Most limit questions are a single recognition: which tool clears the indeterminate form. Get the standard limits cold and the 0/0 questions become one or two lines.

Concept 1 of 5

What a limit is — and when it exists

Intuition

A limit describes the value a function approaches, not necessarily its value at the point. It exists only when both sides agree — the left-hand limit equals the right-hand limit. Always try direct substitution first; only if that gives an indeterminate form (0/0, ∞/∞, 1^∞, …) do you need a technique.

Definition

$\lim_{x\to a}f(x)=L$ means $f(x)$ gets arbitrarily close to $L$ as $x\to a$ . It exists iff LHL $=$ RHL. Algebra of limits: limits distribute over sums, products, and quotients (when denominators are non-zero). For $n\to\infty$ ratios, the dominant term decides (e.g. $\dfrac{a^n+b^n}{a^{n}}\to$ the larger base's contribution).

Worked example

Evaluate

\lim_{x\to 2}(x^2+3x-1)

The function is a polynomial (continuous), so substitute directly.
$2^2+3(2)-1=4+6-1=9$ .

Answer:

9

Practice this conceptself-check · 4 quick reps

Try it yourself

Does

\lim_{x\to 0}\dfrac{|x|}{x}

exist?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
A limit exists iff?
2.
First thing to try on any limit?
3.
Is $\lim_{x\to a}f(x)$ necessarily $f(a)$ ?
4.
$\lim_{x\to 1}(x^2+1)$ ?

Drill 4 more on what a limit is — and when it exists

Concept 2 of 5

0/0 by factoring, cancelling, and rationalising

Intuition

When substitution gives 0/0, the zero factor is shared by numerator and denominator. Factor and cancel it (often via

x^n-a^n

), or — when surds are involved — rationalise by multiplying by the conjugate to expose the cancelling factor.

Definition

Factor/cancel: use $x^n-a^n=(x-a)(x^{n-1}+\cdots+a^{n-1})$ ; the standard result $\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=n\,a^{n-1}$ .
Rationalise: multiply numerator and denominator by the conjugate of the surd to turn $\sqrt{A}-\sqrt{B}$ into $A-B$ , then cancel.

Worked example

Evaluate

\lim_{x\to 2}\dfrac{x^2-4}{x-2}

Factor: $\dfrac{(x-2)(x+2)}{x-2}=x+2$ for $x\neq 2$ .
Substitute: $2+2=4$ .

Answer:

4

Practice this conceptself-check · 4 quick reps

Try it yourself

Evaluate

\lim_{x\to 1}\dfrac{x^9-1}{x^3-1}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\lim_{x\to a}\dfrac{x^n-a^n}{x-a}=?$
2.
Tool for $\dfrac{\sqrt{x}-\sqrt{a}}{x-a}$ ?
3.
$\lim_{x\to1}\dfrac{x^4-1}{x-1}$ ?
4.
$\lim_{x\to1}\dfrac{x^9-1}{x^3-1}$ ?

From the bank · past-year question

Example 2Limits & ContinuityEASY

What is

\lim_{x\to1}\dfrac{x^{9}-1}{x^{3}-1}

equal to?

[Q100 · Apr · 2023]

Drill 12 more on 0/0 by factoring, cancelling, and rationalising

Concept 3 of 5

The standard limits to memorise

Intuition

A fixed list of limits resolves most trigonometric, exponential, and logarithmic 0/0 forms instantly. Reduce the problem to one of these (often by multiplying/dividing by the right variable) and read off the value.

Definition

Each holds as $x\to 0$ (angles in radians). For a scaled argument, the factor scales too — e.g. $\lim_{x\to0}\dfrac{\sin ax}{x}=a$ .

Limit (as x → 0)	Value
sin x / x	1
tan x / x	1
(1 − cos x) / x²	1/2
log(1 + x) / x	1
(eˣ − 1) / x	1
(aˣ − 1) / x	ln a
(1 + x)^(1/x)	e

Radians only. Scale the argument and the value scales: sin(ax)/x → a.

Practice this conceptself-check · 4 quick reps

Try it yourself

Evaluate

\lim_{x\to 0}\dfrac{10^{\sin x}-1}{\tan x}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\lim_{x\to0}\dfrac{\sin x}{x}$ ?
2.
$\lim_{x\to0}\dfrac{1-\cos x}{x^2}$ ?
3.
$\lim_{x\to0}\dfrac{a^x-1}{x}$ ?
4.
$\lim_{x\to0}\dfrac{\sin 5x}{x}$ ?

From the bank · past-year question

Example 3Limits & ContinuityEASY

What is

\lim_{x\to0}\dfrac{10^{\sin x}-1}{\tan x}

equal to?

[Q73 · Apr · 2026]

Drill 8 more on the standard limits to memorise

Concept 4 of 5

L'Hôpital's rule for 0/0 and ∞/∞

Intuition

When a limit is genuinely 0/0 or ∞/∞ and factoring is awkward, differentiate the top and bottom separately and try the limit again. Repeat if it's still indeterminate.

Definition

If $\lim\dfrac{f}{g}$ is $\tfrac00$ or $\tfrac{\infty}{\infty}$ and $f,g$ are differentiable, then $\lim\dfrac{f}{g}=\lim\dfrac{f'}{g'}$ (provided the latter exists). Only apply it to a true indeterminate form — never to a determinate one. Series expansion ( $e^x=1+x+\tfrac{x^2}{2}+\cdots$ , $\sin x=x-\tfrac{x^3}{6}+\cdots$ ) often does the same job faster.

Worked example

Evaluate

\lim_{x\to 0}\dfrac{\sin x - x}{x^3}

It is 0/0. Differentiate top and bottom: $\dfrac{\cos x-1}{3x^2}$ (still 0/0).
Again: $\dfrac{-\sin x}{6x}\to-\tfrac16$ (using $\sin x/x\to 1$ ).

Answer:

-\tfrac16

Practice this conceptself-check · 4 quick reps

Try it yourself

Evaluate

\lim_{x\to 0}\dfrac{e^x-(1+x)}{x^2}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
L'Hôpital applies to which forms?
2.
After differentiating top & bottom, then?
3.
First-order expansion of $e^x$ ?
4.
Can you use it on $\tfrac{2}{3}$ ?

From the bank · past-year question

Example 4Limits & ContinuityEASY

What is

\lim_{x\to 0}\dfrac{e^x-(1+x)}{x^2}

equal to?

[Q71 · Apr · 2017]

Drill 2 more on l'hôpital's rule for 0/0 and ∞/∞

Concept 5 of 5

The 1^∞ form

Intuition

A limit of the shape

[f(x)]^{g(x)}

where

f\to 1

and

g\to\infty

is the indeterminate

1^\infty

. Take logs (or use the standard shortcut) to convert it to a 0·∞ product, which becomes an ordinary limit in the exponent.

Definition

If $f(x)\to 1$ and $g(x)\to\infty$ , then $\lim [f(x)]^{g(x)}=e^{\,\lim\, g(x)\,[f(x)-1]}$ . Equivalently, $L=e^{\lim g\ln f}$ — take $\ln$ , evaluate the resulting product, then exponentiate.

The 1^∞ shortcut

\lim [f(x)]^{g(x)} = e^{\,\lim\, g(x)\,[f(x)-1]}\quad (f\to 1,\ g\to\infty)

Worked example

Evaluate

\lim_{x\to 0}(1+2x)^{1/x}

Form $1^\infty$ : use $L=e^{\lim g[f-1]}$ with $f-1=2x$ , $g=\tfrac1x$ .
Exponent $=\lim_{x\to0}\tfrac{1}{x}\cdot 2x=2$ .

Answer:

e^2

Practice this conceptself-check · 4 quick reps

Try it yourself

Evaluate

\lim_{x\to\infty}\left(1+\tfrac{3}{x}\right)^{x}

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$\lim[f]^{g}$ with $f\to1,g\to\infty$ equals?
2.
$\lim_{x\to0}(1+x)^{1/x}$ ?
3.
$\lim_{x\to\infty}(1+\tfrac2x)^{x}$ ?
4.
What indeterminate form is this?

From the bank · past-year question

Example 5Limits & ContinuityHARD

Let

f(x)=a^{x-1}+b^{x-1}

and

g(x)=\sqrt{x-1}

What is

\lim_{x\to1}f(x)^{1/g(x)}

equal to?

[Q54 · Sep · 2023]

Drill 1 more on the 1^∞ form

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

The 1^∞ form
The 1^∞ shortcut
$\lim [f(x)]^{g(x)} = e^{\,\lim\, g(x)\,[f(x)-1]}\quad (f\to 1,\ g\to\infty)$

Reference tables (1)

The standard limits to memorise7 rows

Limit (as x → 0)	Value
sin x / x	1
tan x / x	1
(1 − cos x) / x²	1/2
log(1 + x) / x	1
(eˣ − 1) / x	1
(aˣ − 1) / x	ln a
(1 + x)^(1/x)	e

Radians only. Scale the argument and the value scales: sin(ax)/x → a.

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Limits & ContinuityEASY

Consider the following statements: 1. If

\lim_{x\to a}f(x)

and

\lim_{x\to a}g(x)

both exist, then

\lim_{x\to a}\{f(x)g(x)\}

exists. 2. If

\lim_{x\to a}\{f(x)g(x)\}

exists, then both

\lim_{x\to a}f(x)

and

\lim_{x\to a}g(x)

must exist. Which of the above statements is/are correct?

[Q93 · Apr · 2017]

Example 2Limits & ContinuityEASY

What is

\lim_{x\to -1}\frac{x^3+x^2}{x^2+3x+2}

equal to?

[Q79 · Apr · 2021]

Example 3Limits & ContinuityEASY

What is the value of

\lim_{x \to 0} \dfrac{\sin x^{\circ}}{\tan 3x^{\circ}}

[Q73 · Sep · 2019]

Example 4Limits & ContinuityHARD

\lim_{x\to a}\frac{a^x-x^a}{x^a-a^a}=-1

, then what is the value of

a

[Q76 · Apr · 2021]

Example 5Limits & ContinuityHARD

Let

f(x)=a^{x-1}+b^{x-1}

and

g(x)=\sqrt{x-1}

What is

\lim_{x\to1}\frac{f(x)-1}{g(x)}

equal to?

[Q53 · Sep · 2023]

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