Precalculus

College and School of Mathematics
Math::Arithmetic::Grade III::Fractions

In the first grade, you master all the whole numbers from 0 to 10 thousand or 1 million. In the second grade, you master addition, subtraction, multiplication, and division of these whole numbers. Just when you thought you know everything about numbers, in the third grade came along fractions.

What are fractions? Well, to begin with, there are more than one kind of numbers. There are three kinds of numbers and they are the following.

WHOLE NUMBERS are all the numbers that you know, 0 apple, 1 apple, 2 apples, 3 apples, and so and on.

FRACTIONS are the numbers that are between the WHOLE NUMBERS. Fractions are less than a whole number. If you cut an apple in half, then you don't have 1 apple. You have two half, or halves, apples. Half is written in numbers as 1/2. The / symbol means, in this case, there is 1 of 2 parts of an apple. You got 1/2 of an apple and the other 1/2 you gave to a friend or your saving it for later.

MIXED NUMBERS are numbers that have WHOLE NUMBERS and FRACTIONS together.

How tall are you? You might say 5 feet and 6 inches. There are two numbers there, 5 and 6. You can say 5 feet and 6 inches in one number only! That would be 5 1/2 feet. The 5 is the WHOLE NUMBER and the 1/2 is the FRACTION.

Write the following numbers and expression as fractions or mixed numbers.

1. One half
2. One third
3. Six and a half
4. 10 feet and 6 inches
5. 1 + 1/10

To infinity and beyond!

John Sindayen

Math::Precalculus::Limits

Calculus problems are based on functions but calculus solutions are based on the idea of limits. What is a limit?

Well, to solve any calculus problem, your equation needs to be expressed as a function, like y = x + 1, where the other side of the equation needs to be one variable y with 1 coefficient or f(x). If the function has a real X value at a certain point, then it also has a real Y value at that given point.

However, if the function does not have a real X value at a given point in the line, then you need to use the concept of limits to come to a value of Y at that given point in the line. Since X has no real value, then we will have to say while the function approaches at this X value rather than saying when the function is at this X value. Why? Because the function never had a real X value at that given point of the function.

Find the limits of the following equations.

1. x = 10 when x approaches 10
2. y = 20 when x approaches 20
3. y = 3x - 4 when x approaches 3
4. 3x = y + 4 when x approaches 2
5. f(x) = sin(1/x) when x approaches 0

Welcome to The Outer Limits!

John Sindayen

Math References:
http://www.mathwarehouse.com/calculus/limits/how-to-determine-when-limits-do-not-exist.php
https://en.wikipedia.org/wiki/The_Outer_Limits_(1963_TV_series)