Limits: Standard Limits and L'Hospital's Rule | NIMCET

Quick answer: $\lim_{x \to a} f(x) = L$ means f(x) gets arbitrarily close to L as x approaches a. The most used standard limits are $\lim_{x \to 0}\frac{\sin x}{x} = 1$ , $\lim_{x \to 0}\frac{e^x - 1}{x} = 1$ , $\lim_{x \to 0}\frac{\ln(1 + x)}{x} = 1$ , $\lim_{x \to a}\frac{x^n - a^n}{x - a} = na^{n - 1}$ and $\lim_{x \to 0}(1 + x)^{\frac{1}{x}} = e$ . For $\frac{0}{0}$ or $\frac{\infty}{\infty}$ forms, L'Hospital's rule gives $\lim\frac{f(x)}{g(x)} = \lim\frac{f'(x)}{g'(x)}$ .

Algebra of limits

If $\lim f(x) = L$ and $\lim g(x) = M$ :

Operation	Limit
Sum / difference	$L \pm M$
Product	$LM$
Quotient	$\frac{L}{M}$ , if $M \ne 0$
Constant multiple	$kL$
Power	$L^n$

Standard limits

Limit	Value
$\lim_{x \to 0}\frac{\sin x}{x}$	1
$\lim_{x \to 0}\frac{\tan x}{x}$	1
$\lim_{x \to 0}\frac{1 - \cos x}{x^2}$	$\frac{1}{2}$
$\lim_{x \to 0}\frac{\sin^{-1}x}{x}$	1
$\lim_{x \to 0}\frac{e^x - 1}{x}$	1
$\lim_{x \to 0}\frac{a^x - 1}{x}$	$\ln a$
$\lim_{x \to 0}\frac{\ln(1 + x)}{x}$	1
$\lim_{x \to a}\frac{x^n - a^n}{x - a}$	$na^{n - 1}$
$\lim_{x \to 0}(1 + x)^{\frac{1}{x}}$	$e$
$\lim_{x \to \infty}\left(1 + \frac{1}{x}\right)^x$	$e$

Angles are in radians.

Indeterminate forms

The forms $\frac{0}{0}$ , $\frac{\infty}{\infty}$ , $0 \cdot \infty$ , $\infty - \infty$ , $1^\infty$ , $0^0$ and $\infty^0$ need extra work.

Method	Use when
Factorise and cancel	Polynomial $\frac{0}{0}$
Rationalise	Square roots in $\frac{0}{0}$
Standard limits	Trig, exponential, log expressions
L'Hospital's rule	$\frac{0}{0}$ or $\frac{\infty}{\infty}$
$1^\infty$ formula	$\lim f^g$ with $f \to 1$ , $g \to \infty$

The $1^\infty$ formula: if $f(x) \to 1$ and $g(x) \to \infty$ , then

$\lim f(x)^{g(x)} = e^{\lim g(x)[f(x) - 1]}$

L'Hospital's rule

If $\frac{f(x)}{g(x)}$ gives $\frac{0}{0}$ or $\frac{\infty}{\infty}$ at a, then

$\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}$

provided the right side exists. Differentiate numerator and denominator separately (not as a quotient), and repeat if the form is still indeterminate.

Worked examples

Example 1: factorisation

$\lim_{x \to 2}\frac{x^2 - 4}{x - 2} = \lim_{x \to 2}(x + 2) = 4$

Example 2: rationalisation

$\lim_{x \to 0}\frac{\sqrt{1 + x} - 1}{x} = \lim_{x \to 0}\frac{1}{\sqrt{1 + x} + 1} = \frac{1}{2}$

Example 3: standard trig limit

$\lim_{x \to 0}\frac{\sin 5x}{\sin 3x} = \lim_{x \to 0}\frac{\sin 5x}{5x} \cdot \frac{3x}{\sin 3x} \cdot \frac{5}{3} = \frac{5}{3}$

Example 4: L'Hospital's rule

$\lim_{x \to 0}\frac{x - \sin x}{x^3} = \lim_{x \to 0}\frac{1 - \cos x}{3x^2} = \frac{1}{3} \cdot \frac{1}{2} = \frac{1}{6}$

Example 5: 1 to the power infinity

Find $\lim_{x \to \infty}\left(1 + \frac{3}{x}\right)^{2x}$ .

$e^{\lim 2x \cdot \frac{3}{x}} = e^6$

Summary table

Idea	Key result
$\frac{\sin x}{x}$	1
$\frac{1 - \cos x}{x^2}$	$\frac{1}{2}$
$\frac{x^n - a^n}{x - a}$	$na^{n - 1}$
$1^\infty$ form	$e^{\lim g(f - 1)}$
L'Hospital	Differentiate top and bottom separately

Exam tips

Substitute first; only use special methods if you get an indeterminate form.
For trig limits, create $\frac{\sin(kx)}{kx}$ by multiplying and dividing.
Series expansions ( $\sin x \approx x - \frac{x^3}{6}$ , $e^x \approx 1 + x + \frac{x^2}{2}$ ) are often faster than repeated L'Hospital.

Next, study one-sided limits and limits at infinity, or see the NIMCET Calculus chapter.

Common mistakes

Applying L'Hospital's rule when the form is not $\frac{0}{0}$ or $\frac{\infty}{\infty}$ .
Using the quotient rule instead of differentiating separately.
Using degrees in trig limits.
Treating $1^\infty$ as 1.

Frequently Asked Questions

What is the limit of sin x/x as x approaches 0?

It equals 1, with x in radians.

What is L'Hospital's rule?

For 0/0 or ∞/∞ forms, lim f(x)/g(x) = lim f′(x)/g′(x), if the second limit exists.

What is the limit of (1 + x)^(1/x) as x approaches 0?

It equals e.

How do you evaluate a 1^∞ limit?

Use lim f^g = e^(lim g(f − 1)) when f → 1 and g → ∞.

What is the limit of (xⁿ − aⁿ)/(x − a) as x approaches a?

It equals naⁿ⁻¹.