Limit Laws

Limit Rules in Calculus with Examples

Basic Limit Rules

1. Constant Rule

Rule: \( \lim_{{x \to c}} k = k \)

Example:

\( \lim_{{x \to 4}} 7 = 7 \)

2. Identity Rule

Rule: \( \lim_{{x \to c}} x = c \)

Example:

\( \lim_{{x \to 5}} x = 5 \)

Limits

Guide to Limits in Calculus

Comprehensive guide to Limits in Calculus, designed for students in the MCV4U course.

1. Introduction to Limits

A limit describes the value that a function f(x) approaches as the input x approaches a particular value. Limits are crucial for understanding continuity, derivatives, and integrals in calculus.

Notation:

\[ \lim_{x \to c} f(x) = L \]

This means as \(x\) approaches \(c\), \(f(x)\) approaches \(L\).

2. Types of Limits

a) Finite Limits at Finite Points

If \(f(x)\) approaches a specific value \(L\) as \(x \to c\), the limit exists.

Example:
\[ \lim_{x \to 2} (3x + 1) = 7 \]
Solution: Substitute \(x = 2\): \[ f(2) = 3(2) + 1 = 7 \]

Finding roots of complex numbers

How to Find the \(n\)-th Roots of Complex Numbers

Finding the \(n\)-th root of a complex number involves converting the number to its polar form and using De Moivre’s Theorem. Below is a detailed, step-by-step guide on how to find both fourth roots and cube roots of complex and real numbers with examples.

Steps to Find the \(n\)-th Roots

Let’s break down the process of finding the \(n\)-th roots of a complex number into three main steps:

Step 1: Convert the Complex Number to Polar Form

A complex number \( z = x + yi \) can be written in polar form as:

\[ z = r(\cos \theta + i \sin \theta) \]

Where:

• r is the modulus: \( r = |z| = \sqrt{x^2 + y^2} \)
• \(\theta\) is the argument: \( \theta = \arg(z) \) found using trigonometry based on the quadrant.

2.3 Circles

Equation of a Circle: MPM2D Grade 10 Mathematics

1. Introduction to the Equation of a Circle

A circle is a set of all points in a plane that are at a fixed distance (called the radius) from a fixed point (called the center). The standard form of the equation of a circle depends on the location of its center.

1.1. Equation of a Circle with Center at the Origin

If the center of the circle is at the origin (0,0), the equation of the circle is:

\[ x^2 + y^2 = r^2 \]

Example:

Consider a circle with a radius of 5 units centered at the origin.

The equation is:

\[ x^2 + y^2 = 5^2 \]

\[ x^2 + y^2 = 25 \]

2.2- Length of a Line Segment

2.2 Length of a Line Segment

1. What is the Distance Formula?

The distance formula is used to determine the distance, \(d\), between two points in the Cartesian plane. If the points are \(A(x₁,y₁)\) and \(B(x₂,y₂)\), the distance between them is given by:

\[d = \sqrt{{(x_2- x_1 )^2+ (y_2- y_1 )^2 }}\]

2. How to Find the Length of a Line Segment?

The length of a line segment is the distance between its endpoints. So, you can use the distance formula to find the length of a line segment. If the endpoints of the line segment are \(A(x₁,y₁)\) and \(B(x₂,y₂)\), the length of the line segment is:

\[length = \sqrt{{(x_2- x_1 )^2+ (y_2- y_1 )^2 }}\]

Naming Organic Compounds

Naming Organic Compounds: A Guide for SCH4U Course

Naming Alkanes with and without Branches

Alkanes are hydrocarbons with only single bonds. The general formula for alkanes is \(C_nH_{2n+2}\).

Unbranched alkanes are named based on the number of carbon atoms in the chain: Methane (1 carbon), Ethane (2 carbons), Propane (3 carbons), Butane (4 carbons), Pentane (5 carbons), Hexane (6 carbons), Heptane (7 carbons), Octane (8 carbons), Nonane (9 carbons), Decane (10 carbons).

2.1 Analytical Geometry- Mid point

2.1 Mid-point and Length of a Line Segment

1. What are coordinates of a point in the Cartesian plane?

The Cartesian plane, also known as the coordinate plane, is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). These lines are perpendicular and intersect at the origin, forming four quadrants.

A point in the Cartesian plane is represented by an ordered pair of numbers \((x,y)\), where ‘x’ is the distance from the point to the y-axis (x-coordinate), and ‘y’ is the distance from the point to the x-axis (y-coordinate).

2. What is the midpoint of a line segment?

The midpoint of a line segment is the point that divides the line segment into two equal segments. It is exactly halfway between the endpoints of the line segment.