Calculus I: Functions, Limits, Continuity, and Differentiation – Structured Study Notes

Functions and Limits

The Idea of a Limit

The concept of a limit is foundational in calculus, describing the behavior of a function as its input approaches a particular value. Limits can be estimated numerically (using tables), graphically, or algebraically.

• Left-hand limit: $\lim_{x \to a^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the left.

• Right-hand limit: $\lim_{x \to a^+} f(x)$ is the value $f(x)$ approaches as $x$ approaches $a$ from the right.

• Existence of a limit: $\lim_{x \to a} f(x)$ exists if and only if both one-sided limits exist and are equal.

Example: For $f(x) = \frac{x-9}{\sqrt{x}-3}$ as $x \to 9$, both one-sided limits approach 6, so $\lim_{x \to 9} f(x) = 6$.

Special Numbers: The number $e$ is defined as $\lim_{x \to 0} (1 + x)^{1/x} = e \approx 2.71828$.

Limits and Graphs

• Limits can be determined graphically by observing the value a function approaches as $x$ nears a point.

• If $f(x)$ does not approach a unique value, the limit does not exist.

• Functions may have discontinuities, jumps, or oscillations that prevent the existence of a limit.

Example: $\lim_{x \to 0} \sin(1/x)$ does not exist due to oscillation.

Infinite Limits and Asymptotes

• If $f(x)$ increases or decreases without bound as $x$ approaches $a$, we write $\lim_{x \to a} f(x) = \infty$ or $-\infty$.

• Vertical asymptote: $x = a$ is a vertical asymptote if $f(x)$ approaches $\pm\infty$ as $x \to a$.

• Horizontal asymptote: $y = b$ is a horizontal asymptote if $\lim_{x \to \pm\infty} f(x) = b$.

Example: $\lim_{x \to 0^+} \frac{1}{x} = \infty$; $x=0$ is a vertical asymptote.

Limit Laws

Limits can be computed using algebraic rules:

• $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

• $\lim_{x \to a} [cf(x)] = c \lim_{x \to a} f(x)$

• $\lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$ (if denominator $\neq 0$)

Example: $\lim_{x \to 4} (5x^2 - 9x - 8) = 36$

Techniques for Indeterminate Forms

• Factorization: Used when both numerator and denominator approach zero.

• Rationalization: Multiply numerator and denominator by a conjugate to simplify square roots.

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

The Sandwich (Squeeze) Theorem

If $f(x) \leq g(x) \leq h(x)$ near $a$ and $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} g(x) = L$.

Example: $\lim_{x \to 0} x \sin(1/x) = 0$

Trigonometric Limits

• $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$

• $\lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} = 0$

Continuity of Functions

Continuity at a Point

A function $f$ is continuous at $a$ if:

• $f(a)$ is defined

• $\lim_{x \to a} f(x)$ exists

• $f(a) = \lim_{x \to a} f(x)$

If any of these fail, $f$ is discontinuous at $a$.

Types of Discontinuities

• Removable: Limit exists, but $f(a)$ is undefined or not equal to the limit.

• Jump: Left and right limits exist but are not equal.

• Infinite: Function approaches $\pm\infty$ near $a$.

Continuity on Intervals

• Continuous on $(a, b)$: continuous at every point in the interval.

• Continuous on $[a, b]$: continuous on $(a, b)$, right-continuous at $a$, left-continuous at $b$.

Theorems on Continuity

• Polynomials, rational, root, trigonometric, exponential, logarithmic, and absolute value functions are continuous on their domains.

• If $f$ and $g$ are continuous at $a$, so are $f+g$, $f-g$, $fg$, $cf$, and $f/g$ (if $g(a) \neq 0$).

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = N$.

Application: Guarantees the existence of roots in an interval where the function changes sign.

Differentiation

Definition of the Derivative

The derivative of $f$ at $a$ is:

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

• Alternatively, $f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$

If this limit exists, $f$ is differentiable at $a$.

Geometric Interpretation: Tangents

• The derivative at $a$ gives the slope of the tangent to the curve at $(a, f(a))$.

• The equation of the tangent line: $y = f(a) + f'(a)(x - a)$

Differentiability vs. Continuity

• If $f$ is differentiable at $a$, then $f$ is continuous at $a$.

• The converse is not true: a function can be continuous but not differentiable (e.g., $f(x) = |x|$ at $x=0$).

Derivative as a Function

The derivative $f'(x)$ is itself a function, defined wherever $f$ is differentiable.

Derivative Rules

• Constant Rule: $\frac{d}{dx} c = 0$

• Constant Multiple Rule: $\frac{d}{dx} [cf(x)] = c f'(x)$

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$ for $n \in \mathbb{R}$

• Sum/Difference Rule: $\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)$

• Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x))g'(x)$

Trigonometric Derivatives

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} \tan x = \sec^2 x$

• $\frac{d}{dx} \csc x = -\csc x \cot x$

• $\frac{d}{dx} \sec x = \sec x \tan x$

• $\frac{d}{dx} \cot x = -\csc^2 x$

Implicit Differentiation

Used when $y$ is defined implicitly as a function of $x$ (e.g., $x^2 + y^2 = 4$). Differentiate both sides with respect to $x$, treating $y$ as a function of $x$ and applying the Chain Rule.

Example: $x^2 + y^2 = 4 \implies 2x + 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$

Derivatives of Exponential and Logarithmic Functions

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} a^x = a^x \ln a$

• $\frac{d}{dx} \ln x = \frac{1}{x}$

• $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$

Higher Order Derivatives

Successive derivatives are denoted $f''(x)$, $f'''(x)$, $f^{(n)}(x)$, etc. The $n$th derivative is $\frac{d^n y}{dx^n}$.

Inverse Functions and Their Derivatives

If $f$ is invertible and differentiable, then $\left(f^{-1}\right)'(a) = \frac{1}{f'(f^{-1}(a))}$.

Derivatives of Inverse Trigonometric Functions

• $\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$

• $\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}}$

• $\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}$

Logarithmic Differentiation

Useful for differentiating complicated products, quotients, or powers. Take the natural logarithm of both sides, differentiate implicitly, and solve for $y'$.

Applications of Differentiation

l'Hôpital's Rule

Used to evaluate limits of indeterminate forms $\frac{0}{0}$ or $\frac{\infty}{\infty}$:

• If $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the latter limit exists).

Related Rates

Problems involving rates at which related variables change. Use implicit differentiation with respect to time $t$.

Example: For a ladder sliding down a wall, relate $x$ and $y$ via $x^2 + y^2 = L^2$, differentiate with respect to $t$ to find $\frac{dy}{dt}$ in terms of $\frac{dx}{dt}$.

Extrema and the Extreme Value Theorem

• Absolute (global) maximum/minimum: The largest/smallest value of $f$ on its domain.

• Local (relative) maximum/minimum: The largest/smallest value of $f$ in a neighborhood.

• Critical number: $c$ where $f'(c) = 0$ or $f'(c)$ does not exist.

• Extreme Value Theorem: If $f$ is continuous on $[a, b]$, $f$ attains both an absolute maximum and minimum on $[a, b]$.

Rolle's Theorem and the Mean Value Theorem (MVT)

• Rolle's Theorem: If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there exists $c \in (a, b)$ with $f'(c) = 0$.

• MVT: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then $\exists c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

Curve Sketching

• Find domain, intercepts, symmetry, asymptotes.

• Use $f'(x)$ to find intervals of increase/decrease and local extrema (First Derivative Test).

• Use $f''(x)$ to find concavity and points of inflection (Second Derivative Test).

• Combine all information to sketch the graph.

Optimization

Optimization problems involve finding the maximum or minimum value of a function subject to constraints. The general strategy is:

1. Define the function to be optimized.

2. Express it in terms of a single variable using constraints.

3. Find critical points and endpoints.

4. Evaluate the function at these points to determine the optimum.

Introduction to Integration

Antiderivatives and Indefinite Integrals

• An antiderivative $F$ of $f$ satisfies $F'(x) = f(x)$.

• The indefinite integral $\int f(x) dx = F(x) + C$ represents all antiderivatives of $f$.

• Power Rule for Integration: $\int x^r dx = \frac{1}{r+1} x^{r+1} + C$ for $r \neq -1$

• $\int \frac{1}{x} dx = \ln|x| + C$

Definite Integrals and the Fundamental Theorem of Calculus (FTC)

• The definite integral $\int_a^b f(x) dx$ is the limit of Riemann sums and represents the net area under $f$ from $a$ to $b$.

• FTC2: If $F$ is any antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$.

Substitution Rule

Used to integrate composite functions. If $u = g(x)$, then $\int f(g(x))g'(x) dx = \int f(u) du$.

Integration by Parts

Based on the product rule for differentiation. If $u = f(x)$ and $dv = g'(x) dx$, then $\int u dv = uv - \int v du$.

Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, but $f(a)$ is undefined or not equal to the limit	$f(x) = \frac{x^2 - 1}{x - 1}$ at $x=1$
Jump	Left and right limits exist but are not equal	Piecewise function with different values on each side
Infinite	Function approaches $\pm\infty$ near $a$	$f(x) = \frac{1}{x}$ at $x=0$

Table: Derivative Rules

Function $y$	Derivative $\frac{dy}{dx}$
Constant $c$	$0$
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$1/x$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$

Table: Indefinite Integrals

Function $f(x)$	Indefinite Integral $\int f(x) dx$
$x^n$ ($n \neq -1$)	$\frac{1}{n+1} x^{n+1} + C$
$\frac{1}{x}$	$\ln|x| + C$
$e^x$	$e^x + C$
$a^x$	$\frac{a^x}{\ln a} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$

Additional info:

• Some advanced topics (Taylor polynomials, hyperbolic functions, etc.) are introduced but not detailed here; see referenced textbook sections for further study.

• For all theorems and rules, proofs are available in standard calculus textbooks and are often required for exams.