Limit ☰

The limit of a function is a fundamental concept in calculus. Informally, it describes the value a function approaches as the input approaches a specific point. Mathematically, the limit of a function \(f(x)\) as \(x\) approaches a point a is denoted as:
\( \underset{x \to a}{\lim} f(x) = L \)
This means that as \(x\) gets arbitrarily close to a, the function values \(f(x)\) get arbitrarily close to \(L\).

Epsilon-Delta Definition of a Limit
The epsilon-delta definition is a formal, rigorous definition of a limit. It states that for every \( \epsilon > 0 \), there exists a \( \epsilon > 0 \) such that if \( 0 < |x–a| < \epsilon \), then \( |f(x)–L| < \epsilon \). This definition captures the idea that as \(x\) gets arbitrarily close to a, the function values \(f(x)\) get arbitrarily close to \(L\).

• The limit of a constant function \(c\) is the constant itself: $$ \underset{x \to a}{\lim} c = c $$
• The limit of a linear function $$ f(x)=mx+b $$ is $$ \underset{x \to a}{\lim} (mx + b) = ma + b $$
• The sum/difference law: $$ \underset{x \to a}{\lim} [f(x) \pm g(x)] = \underset{x \to a}{\lim} f(x) \pm \underset{x \to a}{\lim} g(x) $$
• The product law: $$ \underset{x \to a}{\lim} [f(x) \cdot g(x)] = \underset{x \to a}{\lim} f(x) \cdot \underset{x \to a}{\lim} g(x) $$
• The quotient law: $$ \underset{x \to a}{\lim} \frac{f(x)}{g(x)} = \frac{\underset{x \to a}{\lim} f(x)}{\underset{x \to a}{\lim} g(x)}, \quad \underset{x \to a}{\lim} g(x) \neq 0 $$

One-Sided Limits ☰

One-sided limits consider the function's behavior as the input approaches a point from one side only:

Left-hand limit: $$ \underset{x \to a^-}{\lim} f(x) = L_- $$ Right-hand limit: $$ \underset{x \to a^+}{\lim} f(x) = L_+ $$ If the left-hand and right-hand limits exist and are equal, the overall limit exists, and $$ \underset{x \to a}{\lim} f(x) = L_- = L_+ $$

Limits Involving Infinity ☰

Limits involving infinity can describe the behavior of a function as the input or output approaches infinity. Two common cases are:

1. As \(x\) approaches infinity: $$ \underset{x \to \infty}{\lim} f(x) $$ 2. As f(x) approaches infinity: $$ \underset{x \to a^-}{\lim} f(x) = \infty $$ $$ \underset{x \to a}{\lim} f(x) = +\infty \quad \text{and} \quad \underset{x \to a}{\lim} g(x) = -\infty $$ $$ \underset{x \to a^+}{\lim} f(x) = +\infty \quad \text{and} \quad \underset{x \to a^+}{\lim} g(x) = -\infty $$ $$ \underset{x \to a^-}{\lim} f(x) = +\infty \quad \text{and} \quad \underset{x \to a^-}{\lim} g(x) = -\infty $$ A straight line \(x=a\) is a vertical asymptote of the function \(f(x)\) if any of these relations holds. $$ \underset{x \to +\infty}{\lim} f(x) = b $$ or $$ \underset{x \to -\infty}{\lim} f(x) = b $$ If these limits exist, the straight line \(y=b\) is the horizontal asymptote of the function \(f(x)\).

Continuity ☰

A function is continuous at a point a if the following three conditions are met:

1. \(f(a) \) is defined,

2. \( \underset{x \to a}{\lim} f(x) \) exists,

3. \( \underset{x \to a}{\lim} f(x) = f(a) \)


If a function is continuous at every point in its domain, it is called a continuous function. Continuity has several important properties and implications, such as:

• The sum, difference, product, and quotient of continuous functions are continuous, provided the denominator is nonzero.
• Polynomial and rational functions are continuous on their domains.
• The composition of continuous functions is continuous.
• The Intermediate Value Theorem states that if a continuous function \(f\) takes on values \(f(a)\) and \(f(b)\) for some interval \([a,b] \), then it takes on every value between \(f(a)\) and \(f(b)\) at least once in the interval.

Limit of Trigonometric Functions ☰

Trigonometric functions, such as sine, cosine, and tangent, also have limits. Some important trigonometric limits include: $$ \underset{x \to 0}{\lim} \frac{\sin x}{x} = 1 $$ $$ \underset{x \to 0}{\lim} \frac{1 - \cos x}{x} = 0 $$ $$ \underset{x \to 0}{\lim} \frac{\tan x}{x} = 1 $$

Limit of Exponential and Logarithmic Functions ☰

Exponential and logarithmic functions have important limits as well. Some notable limits are: $$ \underset{x \to 0}{\lim} \frac{e^x - 1}{x} = 1 $$ where \(e\) is the base of the natural logarithm. $$ \underset{x \to 0}{\lim} \frac{\ln(1 + x)}{x} = 1 $$ where \( \ln\) is the natural logarithm.

Limit of a Sequence ☰

A sequence is an ordered list of numbers, often denoted as \(a_n\). The limit of a sequence as \(n\) approaches infinity is defined as: $$ \underset{n \to \infty }{\lim} a_n=a $$ If the terms of the sequence get arbitrarily close to \(L\) as \(n\) increases, then the sequence converges to \(L\). Otherwise, the sequence diverges.

Taylor and Maclaurin Series ☰

Taylor series and Maclaurin series are infinite series representations of a function near a specific point. The Taylor series of a function \(f(x)\) about a point a is given by: $$ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n $$ The Maclaurin series is a special case of the Taylor series, with \(a=0\): $$ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} (x)^n $$

Derivatives and Integrals ☰

Limits are the foundation of derivatives and integrals in calculus. The derivative of a function \(f(x)\) at a point a represents the instantaneous rate of change of the function at that point and is given by the limit: $$ f'(a) = \underset{h \to 0}{\lim} \frac{f(h+a) - f(a)}{h} $$ Similarly, the integral of a function calculates the accumulated change or the area under the curve, and it is defined using limits in the form of the Riemann integral or the more general Lebesgue integral.