Left and Right hand Limit

Mastering Left-Hand and Right-Hand Limits: A Step-by-Step Guide with Examples

Understanding limits is a fundamental concept in calculus. Evaluating left-hand limits (\( \lim_{x \to c^-} \)) and right-hand limits (\( \lim_{x \to c^+} \)) is essential to determine if a limit exists at a given point. This guide walks you through the steps to evaluate these limits and explains when they are equal or unequal, supported by examples to solidify your understanding.

How to Evaluate Left and Right Limits: Step-by-Step Guide

Step 1: Understand the Function and Point of Interest (\(c\))

Identify the function \(f(x)\) and the value \(c\) at which you want to evaluate the limit. Determine if \(f(x)\) is continuous, piecewise-defined, or has discontinuities near \(c\).

Step 2: Find the Left-Hand Limit (\( \lim_{x \to c^-} f(x) \))

Approach the value of \(c\) from the left (values less than \(c\)). Substitute \(x\)-values slightly less than \(c\) into \(f(x)\) and observe the behavior.

Step 3: Find the Right-Hand Limit (\( \lim_{x \to c^+} f(x) \))

Approach \(c\) from the right (values greater than \(c\)). Substitute \(x\)-values slightly greater than \(c\) into \(f(x)\) and observe the behavior.

Step 4: Compare the Two Limits

If \( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \), the limit exists, and \(\lim_{x \to c} f(x)\) equals this common value. If \( \lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x) \), the limit does not exist.

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.

1.6 System of Linear Equations

1.6 Solving Systems of Equations by Elimination Method

The method of elimination is a technique used to solve systems of linear equations. This method involves adding or subtracting the equations to eliminate one of the variables. Here’s a step-by-step process:

Step 1: Write Down the System of Equations

Write down the system of equations that you want to solve. For example, let’s consider the following system of equations:

\(2x + 3y = 12\)

\(4x - 3y = 6\)

1.5 System of equivalent Equations

1.5 Equivalent Equations

1.5.1) Equivalent Equations with Two Variables

Equivalent equations with two variables are equations that have the same set of solutions. For example, the equations \(2x + 3y = 6\) and \(4x + 6y = 12\) are equivalent because they represent the same line in the coordinate plane, and thus have the same set of solutions.

1.4 Solving system of linear equations by substitution

1.4 Solving by Substitution

To solve a system of linear equations by using substitution includes the following steps.

Consider we have system of equations as:

\(x + y = 10\) and \(2x + 3y = 8\)

Step 1: Isolate one of the variables in one of the equations

Choose one of the equations (any of the variables having coefficient as ‘1’) and solve it for one variable in terms of the other. This can be either x or y. For example, if you have the equation \(x + y = 10\), you can isolate \(x\) by subtracting \(y\) from both sides to get \(x = 10 - y\).

Quantum Numbers and Electronic Configuration of atoms

Writing Electronic Configuration of an Atom

1. Identify the atomic number (\(Z\)) of the atom. The atomic number is equal to the number of protons, which is also equal to the number of electrons in a neutral atom.
2. Fill the energy levels (shells) in order of increasing energy. Each energy level is designated by a principal quantum number (\(n\)) starting from 1, 2, 3, and so on.

3. Within each energy level, fill the subshells (\(s,p,d,f\)) in order of increasing energy. Each subshell is designated by an azimuthal quantum number (\(l\)) where \(s=0,p=1,d=2,and f=3\).
4. Each subshell can hold a certain number of electrons: \(s\) can hold 2, \(p\) can hold 6, \(d\) can hold 10, and \(f\) can hold 14.

1.3 Solving System of Equations by Graphing- U1L3MPM2D

1.3 Solving System of Equations by Graphing

Example: Solve the System of Equations by Graphing

Consider the following system of equations:

y = 2x + 4
y = -x + 1

Step 1: Identify the \(y-Intercept\) and Slope

Compare the given equations with the general slope-intercept form of a linear equation, \(y=mx+b\); '\(m\)' is the slope and '\(b\)' is the \(y-intercept\).

For the first equation, the \(y-intercept\) '\(b\)' is 4 and the slope '\(m\)' is 2. This means when \(x = 0\), \(y = 4\), and for every 1 unit increase in \(x\), \(y\) increases by 2 units.

For the second equation, the \(y-intercept\) ‘\(b\)’ is 1 and the slope '\(m\)' is -1. This means when \(x = 0\), y\( = 1\), and for every 1 unit increase in \(x, y\) decreases by 1 unit.

1.2 Graphing Linear System of Equations- U1L2MPM2D

Graphing Linear System of Equations

1) Graphing a Line

A line can be graphed by using following three methods:

• Using table of values
• Using slope and \(y-intercept\)
• Using \(x\) and \(y-intercepts\)

1.1 Linear Equations- U1L1MPM2D

1.1 Solving Linear Equations

Step 1: Simplify Both Sides of the Equation

Before you start solving for the variable, make sure the equation is simplified as much as possible. This means:

• Combine like terms on each side of the equation.
• Apply the distributive property if necessary (i.e., expand brackets).

Step 2: Move Variable Terms to One Side

The goal is to isolate the variable on one side of the equation. If there are terms with the variable on both sides of the equation, use addition or subtraction to move all variable terms to one side.

Step 3: Move Constant Terms to the Opposite Side

Use addition or subtraction to move all constant terms (numbers without variables) to the opposite side of the equation from the variable.

Reciprocal Functions

Reciprocal of Linear and Quadratic Functions

The reciprocal of a function \(f(x)\) is given by \(g(x) = \frac{1}{f(x)}\).

Reciprocal of a Linear Function

Consider a linear function \(f(x) = mx + b\). The reciprocal is \(g(x) = \frac{1}{(mx + b)}\).

Graphing:

• If \(f(x) = 0\), then g(x) is undefined, resulting in a vertical asymptote.
• If \(f(x)\) is positive, then g(x) is also positive.
• If \(f(x)\) is negative, then g(x) is also negative.

Example

Let’s graph the reciprocal of the function \(f(x) = 2x + 1\).

• Identify the zeros of \(f(x): x = \frac{-1}{2}\) (vertical asymptote).
• Determine the sign of \(f(x)\): negative for \(x < \frac{-1}{2}\), positive for \(x > \frac{-1}{2}\).
• Sketch the graph of g(x) using this information.

Sampling and types of Sampling

1) Introduction to Sampling

1.1)            What is sampling?

Welcome to the "Sampling and Types of Sampling" course! In this chapter, we will be exploring the concept of sampling and its importance in various fields of study. Sampling is a fundamental technique used to gather data from a larger population in a systematic and efficient manner.

But what exactly is sampling? Let's break it down:

How to wright news article?

Writing News Articles

Writing a news article involves addressing the five W's and one H: Who, What, When, Where, Why, and How. These elements provide a comprehensive and structured way to convey information. Here's a breakdown of each:

1. Who:
   • Identify the key individuals or groups involved in the news story.
   • Include names, titles, and any relevant background information.
   • Answer questions like: Who is the main subject of the story? Who is affected by the events?