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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 \(L\).

Limits have several important properties, such as:

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 \(f(x)\).


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:

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.