Limits and Calculus: From Continuity to Taylor Series

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 $L$.

Limits have several important properties, such as:

• 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.