Limit in Calculus – Definition, Formula & Applications :: Danielitte

Convex Quadrilateral

Segment of Circle

Sector of Circle

Regular Poligon of N Sides

Sperical Segment

Sperical Sector

Frustum of Right Circular Cone

Triangular Prism

Arithmetic Progression

Geometric Progression

Decimal Logarithm

Natural Logarithm

Euler's Formula

Trigonometric Functions For A Right Triangle

Trigonometric Table

Multiple Angle Formulas

Powers Of Trigonometric Functions

Formulas With T=tan(x/2)

Addition Formulas

Sum Of Trigonometric Functions

Product Of Trigonometric Functions

Half Angle Formulas

Angles Of A Plane Triangle

Sides And Angles Of A Plane Triangle

Relationships Among Trigonometric Fuctions

Linear Equation

System of Two Linear Equation

Quadratic Equation

Exponential Equation

Logarithmic Equation

Trigonometric Equation Cos

Trigonometric Equation Sin

Trigonometric Equation Tan

Trigonometric Equation Cotan

Linear Inequation

Quadratic Inequation

Exponential Inequation

Logarithmic Inequation

Trigonometric Inequation Cos

Trigonometric Inequation Sin

Trigonometric Inequation Tan

Trigonometric Inequation Cotan

Horizontal Shifting

Vertical Shifting

Equation of Line

Equation of Circle

Equation of Line Joining Two Points A,B

Equation of Sphere Center at M and Radius R in Rectangular Coordinates

Equation of Ellipsoid With Center M and Semi-Axes A,B,C

Elliptic Cylinder With Axis as Z Axis

Elliptic Cone With Axis as Z Axis

Hyperboloid of One Sheets

Hyperboloid of Two Sheets

Elliptic Paraboloid

Hyperbolic Paraboloid

Differentiation

Indefinite Integrals

Integrals By Partial Functions

Integrals Involving Roots

Integrals Involving Trigonometric Functions

Transformations

Definite Integrals

Transpose Matrix

Addition And Substraction Of Matrices

Multiplication Of Matrices

Determinant Of Matrices

Inverse Of Matrix

Equation In Matrix Form

Properties Of Matrix Calculations

Relative Complement of A in B

Absolute Complement

Symmetric Difference

Operations On Sets

Standard Deviation

Root Mean Square

Normal Distribution(gaussion Distribution)

Exponential Distribution

Poisson Distribution

Uniforn Distribution

Real Form Of Fourier Series

Parseval's Theorem

Fourier Transform

Fourier Symmetry Relationships

Fourier Transform Pairs

Substitution(Frequency Shifting)

Translation(Time Shifting)

Laplace Transform Pairs

Derivative - Limit

Limits

Definition and Standard Limit Forms

A limit describes the value that a function approaches as the input approaches a certain point. Limits are foundational in calculus and help define continuity, derivatives, and integrals.

Basic Properties

\[ \lim(x + y - z) = \lim x + \lim y - \lim z \]
\[ \lim(xyz) = \lim x \cdot \lim y \cdot \lim z \]
\[ \lim\left(\frac{x}{y}\right) = \frac{\lim x}{\lim y} \]
\[ \lim_{x \to \alpha} [cf(x)] = c \cdot \lim_{x \to \alpha} f(x) \]
\[ \lim_{x \to \alpha} [f(x)]^n = \left(\lim_{x \to \alpha} f(x)\right)^n \]

Limits Involving Exponentials and Logarithms

\[ \lim_{x \to \infty} e^x = \infty \quad,\quad \lim_{x \to -\infty} e^x = 0 \]
\[ \lim_{x \to 0} a^x = 1 \]
\[ \lim_{x \to \infty} \ln x = \infty \]
\[ \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a, \quad a > 0 \]
\[ \lim_{x \to 0} \frac{x}{\log_a(1 + x)} = \frac{1}{\log_a e} \]

Limits Approaching Infinity

\[ \lim_{x \to \infty} \frac{c}{x^n} = 0 \quad (n > 0) \]
\[ \lim_{x \to \infty} \frac{x}{x \sqrt{x!}} = e \]
\[ \lim_{x \to \infty} \left(1 + \frac{k}{x}\right)^x = e^k \]
\[ \lim_{x \to \infty} \left(1 - \frac{1}{x}\right)^x = \frac{1}{e} \]
\[ \lim_{x \to \infty} \left(\frac{\sqrt{2\pi x}}{x!}\right)^{1/x} = e \]
\[ \lim_{x \to \infty} \frac{x}{x^{e^{-x}} \sqrt{x}} = \sqrt{2x} \]

Standard Trigonometric and Inverse Limits

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \quad,\quad \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
\[ \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \quad,\quad \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2} \]
\[ \lim_{x \to 0} \frac{\arcsin x}{x} = 1 \quad,\quad \lim_{x \to 0} \frac{\arctan x}{x} = 1 \]
\[ \lim_{x \to 1} (\arccos x)^2 = 2 \]

Terminology

Limit: The value a function approaches as the input approaches a specific value.
Approaches: Getting closer to a number from either direction on the number line.
Indeterminate form: An expression like \( \frac{0}{0} \) or \( \infty - \infty \) where limits require deeper analysis.
Infinity (\( \infty \)): A concept of being unbounded or limitless.
One-sided limits: Limits from only one side (left-hand or right-hand limit).

Applications

Fundamental to calculus: defines derivatives and integrals.
Used in physics to model instantaneous velocity and acceleration.
Essential in economics for calculating marginal cost and profit.
Helps analyze rates of change in biology, chemistry, and population models.
Used in machine learning and numerical methods for convergence analysis.