Limits, Continuity & Differentiability Formula Sheet

One-page rapid revision of every formula, standard result, and test from this chapter. Use it the night before the exam to refresh limits, continuity, differentiability, the differentiation toolkit, and all derivative applications.

Limits: Definition & Basics

Concept	Result	Notes
Limit	$\lim_{x \to a} f(x) = L$	$f(x)$ approaches $L$ as $x \to a$
Existence	$\lim_{x \to a} f(x)$ exists $\iff$ LHL $=$ RHL	First check for piecewise functions
LHL	$\lim_{x \to a^-} f(x)$	Approach from the left
RHL	$\lim_{x \to a^+} f(x)$	Approach from the right

$\varepsilon$-$\delta$ definition: For every $\varepsilon > 0$, there exists $\delta > 0$ such that

$$0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$$

Algebra of Limits

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$:

Property	Formula
Sum / Difference	$\lim [f(x) \pm g(x)] = L \pm M$
Product	$\lim [f(x) \cdot g(x)] = L \cdot M$
Quotient	$\lim \dfrac{f(x)}{g(x)} = \dfrac{L}{M}$ $(M \neq 0)$
Power	$\lim [f(x)]^n = L^n$
Constant multiple	$\lim [c \cdot f(x)] = c \cdot L$

Limits at Infinity (Rational Functions)

For $\lim_{x \to \infty} \dfrac{a_n x^n + \ldots}{b_m x^m + \ldots}$:

Condition	Limit
$n < m$	$0$
$n = m$	$\dfrac{a_n}{b_m}$
$n > m$	$\pm \infty$

Standard Limits

Trigonometric

$$\boxed{\lim_{x \to 0} \frac{\sin x}{x} = 1}$$

Limit	Value
$\lim_{x \to 0} \dfrac{\tan x}{x}$	$1$
$\lim_{x \to 0} \dfrac{\sin^{-1} x}{x}$	$1$
$\lim_{x \to 0} \dfrac{\tan^{-1} x}{x}$	$1$
$\lim_{x \to 0} \dfrac{1 - \cos x}{x^2}$	$\dfrac{1}{2}$
$\lim_{x \to 0} \dfrac{1 - \cos x}{x}$	$0$
$\lim_{x \to 0} \dfrac{\sin(ax)}{x}$	$a$
$\lim_{x \to 0} \dfrac{\sin(ax)}{\sin(bx)}$	$\dfrac{a}{b}$
$\lim_{x \to 0} \dfrac{\tan(ax)}{\tan(bx)}$	$\dfrac{a}{b}$
$\lim_{x \to 0} \dfrac{1 - \cos(ax)}{x^2}$	$\dfrac{a^2}{2}$

Always in radians, never degrees.

Exponential & Logarithmic

$$\boxed{\lim_{x \to 0} \frac{e^x - 1}{x} = 1} \qquad \boxed{\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1}$$

Limit	Value
$\lim_{x \to 0} \dfrac{a^x - 1}{x}$	$\ln a$
$\lim_{x \to 0} \dfrac{a^x - b^x}{x}$	$\ln\!\left(\dfrac{a}{b}\right)$
$\lim_{x \to 0} \dfrac{\log_a(1 + x)}{x}$	$\dfrac{1}{\ln a}$
$\lim_{x \to 0} \dfrac{\ln(1 + ax)}{x}$	$a$

The “Magic $e$” Family

$$\boxed{\lim_{x \to 0} (1 + x)^{1/x} = e} \qquad \boxed{\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e}$$$$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a \qquad \lim_{x \to a} \left(1 + \frac{1}{f(x)}\right)^{f(x)} = e \;\; (f(x) \to \infty)$$

Algebraic (Power) Limits

$$\boxed{\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}} \qquad \lim_{x \to 0} \frac{(1 + x)^n - 1}{x} = n$$

Indeterminate Forms & L’Hopital

The seven indeterminate forms: $\dfrac{0}{0}$, $\dfrac{\infty}{\infty}$, $0 \times \infty$, $\infty - \infty$, $1^\infty$, $0^0$, $\infty^0$.

$\dfrac{1}{0}$ (or $\dfrac{c}{0}$, $c \neq 0$) is undefined ($\pm\infty$), not indeterminate.

L’Hopital’s Rule (for $\dfrac{0}{0}$ or $\dfrac{\infty}{\infty}$ only):

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

Differentiate numerator and denominator separately (not the quotient rule).

Form	Technique
$\dfrac{0}{0}$	Factor / rationalize / L’Hopital
$\dfrac{\infty}{\infty}$	Divide by highest power / L’Hopital
$0 \cdot \infty$	Rewrite as $\dfrac{\ln x}{1/x}$ type
$\infty - \infty$	Combine into single fraction / rationalize
$1^\infty,\; 0^0,\; \infty^0$	Take $\ln$, apply L’Hopital, then $\lim y = e^{\lim \ln y}$

Continuity

A function $f$ is continuous at $x = a$ when all three (mnemonic DEL) hold:

$$\boxed{\lim_{x \to a} f(x) = f(a)}$$

1. Defined: $f(a)$ exists
2. Limit Exists: LHL $=$ RHL
3. Limit equals value: $\lim_{x \to a} f(x) = f(a)$

On a closed interval $[a, b]$: continuous on $(a, b)$, right-continuous at $a$ $(\lim_{x \to a^+} f = f(a))$, and left-continuous at $b$ $(\lim_{x \to b^-} f = f(b))$.

Types of Discontinuity

Type	Condition	Removable?	Example
Removable	Limit exists but $\neq f(a)$	Yes (redefine $f(a)$)	$\dfrac{x^2-1}{x-1}$ at $x=1$
Jump	LHL $\neq$ RHL	No	$\lfloor x \rfloor$ at integers
Infinite	Limit $= \pm\infty$	No	$\dfrac{1}{x}$ at $x=0$
Oscillatory	Limit DNE (oscillation)	No	$\sin(1/x)$ at $x=0$

Continuity Reference

Function	Continuous?
Polynomials	Everywhere
$\sin x,\ \cos x$	Everywhere
$\tan x,\ \sec x$	Except $\frac{\pi}{2} + n\pi$
$e^x,\ a^x$	Everywhere
$\ln x$	For $x > 0$
$1/x$	Except $x = 0$
$	x
$\lfloor x \rfloor$	Except integers

Sums, differences, products of continuous functions are continuous; quotients where denominator $\neq 0$; compositions of continuous functions are continuous.

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $k$ lies between $f(a)$ and $f(b)$, then $\exists\, c \in (a, b)$ with $f(c) = k$.

Differentiability

Derivative (first principles):

$$\boxed{f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}}$$

LHD / RHD and the test:

$$f'(a^-) = \lim_{h \to 0^-} \frac{f(a+h) - f(a)}{h}, \qquad f'(a^+) = \lim_{h \to 0^+} \frac{f(a+h) - f(a)}{h}$$$$\boxed{f \text{ differentiable at } a \iff f'(a^-) = f'(a^+) \text{ (both finite)}}$$

Key implication chain:

$$\boxed{\text{Differentiable} \implies \text{Continuous} \implies \text{Limit exists}}$$

The converse is false: $f(x) = |x|$ is continuous at $0$ but not differentiable (sharp corner).

NOT differentiable when: sharp corner/cusp, vertical tangent (e.g. $x^{1/3}$ at $0$, derivative $\to \infty$), or any discontinuity.

Basic Derivatives

$f(x)$	$f'(x)$
$c$	$0$
$x^n$	$nx^{n-1}$
$\sqrt{x}$	$\dfrac{1}{2\sqrt{x}}$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$\dfrac{1}{x}$
$\log_a x$	$\dfrac{1}{x \ln a}$

Trigonometric

$f(x)$	$f'(x)$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$

Inverse Trigonometric

$f(x)$	$f'(x)$
$\sin^{-1} x$	$\dfrac{1}{\sqrt{1-x^2}}$
$\cos^{-1} x$	$\dfrac{-1}{\sqrt{1-x^2}}$
$\tan^{-1} x$	$\dfrac{1}{1+x^2}$
$\cot^{-1} x$	$\dfrac{-1}{1+x^2}$
$\sec^{-1} x$	$\dfrac{1}{
$\csc^{-1} x$	$\dfrac{-1}{

Rules of Differentiation

Rule	Formula
Product	$(uv)' = u'v + uv'$
Product (triple)	$(uvw)' = u'vw + uv'w + uvw'$
Quotient	$\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$
Chain	$[f(g(x))]' = f'(g(x)) \cdot g'(x)$
Chain (nested)	$[f(g(h(x)))]' = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$

Pattern Shortcuts

Pattern	Derivative
$(ax + b)^n$	$an(ax + b)^{n-1}$
$e^{ax}$	$ae^{ax}$
$\sin(ax)$	$a\cos(ax)$
$\ln(ax + b)$	$\dfrac{a}{ax + b}$

Logarithmic, Implicit & Parametric

Technique	When	Method / Formula
Logarithmic	Variable exponents ($x^x$, $x^{\sin x}$), big products	Take $\ln$ both sides, then differentiate
Implicit	Equation not solved for $y$	Differentiate both sides w.r.t. $x$; $\dfrac{d}{dx}(y^n) = ny^{n-1}\dfrac{dy}{dx}$
Parametric	$x = f(t),\ y = g(t)$	$\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$

Implicit key patterns:

$$\frac{d}{dx}(y^n) = ny^{n-1}\frac{dy}{dx}, \quad \frac{d}{dx}(xy) = y + x\frac{dy}{dx}, \quad \frac{d}{dx}(\sin y) = \cos y \cdot \frac{dy}{dx}, \quad \frac{d}{dx}(e^y) = e^y\frac{dy}{dx}$$

Higher Order Derivatives

Notation: $f''(x) = \dfrac{d^2y}{dx^2}$ (not $(dy/dx)^2$), up to $f^{(n)}(x) = \dfrac{d^n y}{dx^n}$.

$n$th Derivative Formulas

Function	$n$th derivative
$x^n$	$n!$ (and $0$ for higher orders)
$e^{ax}$	$a^n e^{ax}$
$a^x$	$a^x (\ln a)^n$
$\sin(ax)$	$a^n \sin\!\left(ax + \dfrac{n\pi}{2}\right)$
$\cos(ax)$	$a^n \cos\!\left(ax + \dfrac{n\pi}{2}\right)$
$\ln x$	$\dfrac{(-1)^{n-1}(n-1)!}{x^n}$
$\dfrac{1}{ax+b}$	$\dfrac{(-1)^n \, n! \, a^n}{(ax+b)^{n+1}}$

Leibniz’s Theorem ($n$th derivative of a product)

$$\boxed{(uv)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} u^{(k)} v^{(n-k)}}$$

Same coefficients as the binomial expansion $(a+b)^n$.

Second Derivative — Special Forms

Form	Formula
Parametric	$\dfrac{d^2y}{dx^2} = \dfrac{\frac{d}{dt}\!\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$
Implicit	Differentiate $\dfrac{dy}{dx}$ implicitly again

Parametric second derivative is NOT $\dfrac{d^2y/dt^2}{d^2x/dt^2}$.

Applications of Derivatives

Monotonicity

$$\boxed{f'(x) > 0 \Rightarrow \text{increasing}, \quad f'(x) < 0 \Rightarrow \text{decreasing}, \quad f'(x) = 0 \Rightarrow \text{constant (on interval)}}$$

Maxima & Minima

Critical point: $f'(c) = 0$ or $f'(c)$ does not exist. All local extrema occur at critical points (but not every critical point is an extremum).

Test	Condition	Result
First derivative	$f'$ changes $+ \to -$	Local maximum
	$f'$ changes $- \to +$	Local minimum
	No sign change	Neither (inflection)
Second derivative	$f''(c) < 0$	Local maximum (concave down $\cap$)
	$f''(c) > 0$	Local minimum (concave up $\cup$)
	$f''(c) = 0$	Inconclusive — use first-derivative test

Global extrema on $[a, b]$: evaluate $f$ at all critical points and both endpoints; compare. (Extreme Value Theorem guarantees they exist if $f$ is continuous on $[a, b]$.)

Tangents & Normals

At $(x_0, y_0)$ on $y = f(x)$, with $m_T = f'(x_0)$:

$$\boxed{\text{Tangent: } y - y_0 = f'(x_0)(x - x_0)} \qquad \boxed{\text{Normal: } y - y_0 = -\frac{1}{f'(x_0)}(x - x_0)}$$

Angle between two curves (slopes $m_1, m_2$ at intersection):

$$\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$$

Orthogonal (perpendicular) curves: $m_1 \cdot m_2 = -1$.

Implicit tangent example: $x^2 + y^2 = r^2 \Rightarrow \dfrac{dy}{dx} = -\dfrac{x}{y}$. Polar slope: $\dfrac{dy}{dx} = \dfrac{r'\sin\theta + r\cos\theta}{r'\cos\theta - r\sin\theta}$.

Rate of Change (Related Rates)

Differentiate the relating equation w.r.t. time $t$, then substitute given values.

• Expanding circle: $A = \pi r^2 \Rightarrow \dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt}$
• Sliding ladder: $x^2 + y^2 = L^2 \Rightarrow x\dfrac{dx}{dt} + y\dfrac{dy}{dt} = 0$

Mean Value Theorems

Theorem	Conditions	Conclusion
Rolle’s	Continuous on $[a,b]$, differentiable on $(a,b)$, $f(a)=f(b)$	$\exists c:\ f'(c) = 0$
Lagrange (LMVT)	Continuous on $[a,b]$, differentiable on $(a,b)$	$\exists c:\ f'(c) = \dfrac{f(b)-f(a)}{b-a}$
Cauchy	$f, g$ continuous on $[a,b]$, differentiable on $(a,b)$, $g' \neq 0$	$\exists c:\ \dfrac{f'(c)}{g'(c)} = \dfrac{f(b)-f(a)}{g(b)-g(a)}$

$$\boxed{f'(c) = \frac{f(b) - f(a)}{b - a}}$$

LMVT is a special case of Cauchy’s ($g(x) = x$); Rolle’s is LMVT with $f(a) = f(b)$.

Bounding via LMVT: if $|f'(x)| \le M$ on $[a, b]$, then $|f(b) - f(a)| \le M|b - a|$.

Curve Sketching

Sign Tests

Concept	Test	Meaning
Increasing	$f'(x) > 0$	Rises
Decreasing	$f'(x) < 0$	Falls
Concave up $\cup$	$f''(x) > 0$	Cup
Concave down $\cap$	$f''(x) < 0$	Cap
Inflection point	$f''(x) = 0$ (or undefined) AND $f''$ changes sign	Concavity switches

$f''(c) = 0$ alone is not an inflection point (e.g. $x^4$ at $0$). The sign change is required.

Asymptotes (the “Three D’s”)

Type	Condition
Vertical	Denominator $= 0$ (limit $\to \pm\infty$)
Horizontal	$\lim_{x \to \pm\infty} f(x) = L \Rightarrow y = L$
Oblique	$\deg(\text{num}) = \deg(\text{den}) + 1 \Rightarrow$ long division gives $y = mx + c$

For rational $\dfrac{p(x)}{q(x)}$: $\deg p < \deg q \Rightarrow y = 0$; $\deg p = \deg q \Rightarrow y = \dfrac{\text{lead}(p)}{\text{lead}(q)}$; $\deg p > \deg q \Rightarrow$ no horizontal asymptote.

Symmetry & Polynomials

• Even: $f(-x) = f(x)$ ($y$-axis symmetry). Odd: $f(-x) = -f(x)$ (origin symmetry).
• Degree-$n$ polynomial: at most $n - 1$ turning points. Even degree $\Rightarrow$ same end behavior; odd degree $\Rightarrow$ opposite ends.

Checklist order: Domain $\to$ Intercepts $\to$ Symmetry $\to$ Asymptotes $\to$ $f'$ (monotonicity) $\to$ $f''$ (concavity) $\to$ key points $\to$ sketch.

The Big Picture

graph LR
    A[Limits] --> B[Continuity]
    B --> C[Differentiability]
    C --> D[Differentiation Rules]
    D --> E[Higher Derivatives]
    C --> F[Applications]
    F --> G[Maxima & Minima]
    F --> H[Tangents & Normals]
    F --> I[Rate of Change]
    B --> J[Mean Value Theorems]
    C --> J
    F --> K[Curve Sketching]

Topic Deep-Dives

• Introduction to Limits
• Standard Limits & L’Hopital’s Rule
• Continuity of Functions
• Differentiability & Derivatives
• Rules of Differentiation
• Implicit Differentiation & Parametric Forms
• Higher Order Derivatives
• Applications of Derivatives
• Maxima and Minima
• Mean Value Theorems
• Curve Sketching