Differential Calculus — Limits, Continuity, Derivatives & Mean Value Theorems — Engineering Mathematics Notes

Last reviewed 16 Sept 2026 · 7 min read

Limits

$lim_{x \to a} f (x) = L$ means $f (x)$ approaches $L$ as $x$ approaches $a$ (from both sides). The limit exists iff the left-hand limit (LHL) and right-hand limit (RHL) exist and are equal.

Standard limits

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

Indeterminate forms and L'Hôpital's rule

Forms $\frac{0}{0}$ , $\frac{\infty}{\infty}$ , $0 \cdot \infty$ , $\infty - \infty$ , $1^{\infty}$ , $0^{0}$ , $\infty^{0}$ .

L'Hôpital's rule: for $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ,

lim \frac{f ( x )}{g ( x )} = lim \frac{f ^{'} ( x )}{g ^{'} ( x )}

(repeat if still indeterminate). Other forms are converted: $0 \cdot \infty$ to a quotient; $1^{\infty}$ , $0^{0}$ , $\infty^{0}$ by taking logarithms. For $1^{\infty}$ : $lim f^{g} = e^{l i m g (f - 1)}$ .

Continuity

$f$ is continuous at $x = a$ if $lim_{x \to a} f (x) = f (a)$ (limit exists, function defined, and they are equal).

Discontinuity	Description
Removable	Limit exists but ≠ $f (a)$ or $f (a)$ undefined
Jump	LHL and RHL exist but are unequal
Infinite	Function tends to ±∞
Oscillatory	e.g. $sin (1/ x)$ at 0

Polynomials, $e^{x}$ , $sin x$ , $cos x$ are continuous everywhere; rational functions except where the denominator is zero.

Differentiability

f^{'} (a) = h \to 0 lim \frac{f ( a + h ) - f ( a )}{h}

exists (left and right derivatives equal).

Differentiable ⇒ continuous, but continuous does not imply differentiable — e.g. $f (x) = ∣ x ∣$ is continuous at 0 but not differentiable (corner: left derivative −1, right derivative +1).

Derivatives

Rules

$(u \pm v)^{'} = u^{'} \pm v^{'}$ ; $(c u)^{'} = c u^{'}$
Product: $(uv)^{'} = u^{'} v + u v^{'}$
Quotient: $(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$
Chain rule: $\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$

Standard derivatives

$f (x)$	$f^{'} (x)$	$f (x)$	$f^{'} (x)$
$x^{n}$	$n x^{n - 1}$	$sin x$	$cos x$
$e^{x}$	$e^{x}$	$cos x$	$- sin x$
$a^{x}$	$a^{x} ln a$	$tan x$	$sec^{2} x$
$ln x$	$1/ x$	$sec x$	$sec x tan x$
$lo g_{a} x$	$\frac{1}{x ln a}$	$csc x$	$- csc x cot x$
$sin^{- 1} x$	$\frac{1}{1 - x ^{2}}$	$cot x$	$- csc^{2} x$
$cos^{- 1} x$	$\frac{- 1}{1 - x ^{2}}$	$sinh x$	$cosh x$
$tan^{- 1} x$	$\frac{1}{1 + x ^{2}}$	$cosh x$	$sinh x$

Special techniques

Implicit differentiation — differentiate both sides treating $y$ as a function of $x$ .
Parametric: $x = x (t)$ , $y = y (t)$ → $\frac{d y}{d x} = \frac{d y / d t}{d x / d t}$ ; $\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d t} (\frac{d y}{d x}) / \frac{d x}{d t}$ .
Logarithmic differentiation — for $u^{v}$ or long products: take $ln$ first.

Leibniz theorem (nth derivative of a product)

(uv)_{n} = r = 0 \sum n^{n} C_{r} u_{n - r} v_{r}

Useful nth derivatives: $(e^{a x})_{n} = a^{n} e^{a x}$ ; $(sin a x)_{n} = a^{n} sin (a x + \frac{nπ}{2})$ ; $(\frac{1}{x + a})_{n} = \frac{( - 1 ) ^{n} n !}{( x + a ) ^{n + 1}}$ .

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Differential Calculus — Limits, Continuity, Derivatives & Mean Value Theorems