Understanding Indeterminate Quantities

Introduction to Indeterminate Quantities

• This math branch addresses an issue that previously resulted in math errors, aiming to resolve this problem.

Definite vs. Indeterminate Quantities

• Definite Quantity: Easily determined and calculable; a specific value.
  • Example: 5 divided by 2 is a definite quantity because its value can be precisely determined
• Indeterminate Quantity: A quantity with multiple possible values, making it difficult to specify a single answer.
  • Defined as zero divided by zero
  • Several potential results due to multiple possibilities
  • Illustrated by the equation 0/0 = x, where x could be 1, 2, 3, etc., since 0 times any of these numbers equals 0

Understanding Division and Indeterminate Forms

• Division is the inverse operation of multiplication.
• For example, 5/1 = 5 because 1 * 5 = 5.
• The problem with 0/0 arises because there are multiple numbers that, when multiplied by 0, result in 0, making it impossible to determine a unique value.

Undefined Quantities

• Undefined Quantity: Any non-zero number divided by zero
  • Different from indeterminate form of zero divided by zero
  • A number divided by zero results in infinity, is undefined.
  • Example: a/0, where a is any number other than zero
  • The branch focuses on resolving issues with the indeterminate form 0/0.

Addressing 0/0 Problems

• The math branch aims to solve or provide methods to work around the indeterminate form of 0/0.
• The solution is not always guaranteed.

Example of an Indeterminate Form Problem

• Consider the function (x^2 - 1) / (x - 1).
• Substituting x = 1 directly into the function results in (1^2 - 1) / (1 - 1) = 0/0.

Approach to Solving Indeterminate Forms

• Since direct substitution leads produces the indeterminate form
• Choose numbers slightly less than and slightly greater than x=1
• By calculating the function's value at these points, a trend can be identified, allowing an approximate solution to be found for the indeterminate form.
  • This involves evaluating the function at values close to 1 on both sides (left and right).
  • For example, test values like 0.9, 0.99, 1.1, and 1.01 in the function (x^2 - 1) / (x - 1).### Approaching Limits
• Limits involve examining the behavior of a function as the input (x) gets close to a particular value.
• When direct substitution results in an indeterminate form (0/0), techniques are needed to evaluate the limit.
• The text illustrates approaching a value, such as 1, from both the left and the right sides.

Left and Right Limits

• To find the limit at a specific x-value, examine the function's behavior as x approaches that value from the left (values less than x) and from the right (values greater than x).
• Approaching from the left is denoted as "x approaches c from the left".
• Approaching from the right is denoted as "x approaches c from the right".

Notation

• lim (x→c) f(x) represents the limit of the function f(x) as x approaches c.
• lim (x→c-) f(x) means the limit of f(x) as x approaches c from the left (left-hand limit).
• lim (x→c+) f(x) means the limit of f(x) as x approaches c from the right (right-hand limit).
• For a limit to exist, the left-hand limit and the right-hand limit must be equal.

Evaluating Limits

• If lim (x→c-) f(x) = L and lim (x→c+) f(x) = L, then lim (x→c) f(x) = L.
• If the left-hand limit and the right-hand limit are not equal, then the limit does not exist (DNE).

Graphical Interpretation of Limits

• To determine a limit graphically, approach the x-value from both sides on the graph of the function.
• Observe the y-value that the function approaches from each direction. If they are the same, that is the limit.
• If there is a discontinuity (like a hole) at x=c, the limit may still exist if the left and right limits agree.
• The existence of a limit does not depend on whether the function is defined at that point. A function doesn't need to have a closed dot (defined value) at x=c for the limit to exist.
• When evaluating graphically, move along the graph from the left and right towards the x-value of interest; the y-value where you're heading is the limit.

Examples of Limit Evaluation

• The text provides examples where the limit exists, does not exist, or approaches infinity.
• The text emphasizes that even if a function is undefined at a point, it can still have a limit as x approaches that point.
• Example: Approaching x=0 from the right has the graph going towards 1, while approaching from the left, there's no graph.

Limits at Specific Points

• The text goes through multiple examples to find the limit at x = 0, 1, 2, 3, 4, and 5, demonstrating how to find both right and left limits, and based on that fact knowing if the limit exists.
• When x approaches one from the right side, the graph tends towards two, so f(1) = 2.
• When approaching 1 from the left, the graph tends towards 2, so f(1) = 2.
• Therefore the limit can be written f(x) = 2 where x tends to 1.

Function Definition

• The value of the function (defined with a closed dot) at a specific location is not needed to find the limit.
• For example, at x=2, there exits a closed dot.

Theorem 1: Polynomial and Rational Functions

• If f(x) is a polynomial function, the limit as x approaches c can be found by direct substitution.
• Direct substitution means replacing x with the value c.

Limit Laws

• Limit of a sum/difference: lim [f(x) ± g(x)] = lim f(x) ± lim g(x)
• Limit of a product: lim [f(x) * g(x)] = lim f(x) * lim g(x)
• Limit of a constant times a function: lim [k * f(x)] = k * lim f(x) (k is a constant)
• Limit of a quotient: lim [f(x) / g(x)] = lim f(x) / lim g(x)

Power Rule for Limits

• If n is a rational number, then lim [f(x)]^n = [lim f(x)]^n provided that [lim f(x)]^n is a real number.

Indeterminate Form (0/0) and Factorization

• If direct substitution results in 0/0, algebraic manipulation (like factoring) is needed to simplify the expression.
• The goal is to cancel out the common factor that causes both the numerator and denominator to approach zero.

Canceling Factors to Find Limits

• Factoring and canceling enables evaluating limits.

Conjugate Multiplication and Rationalization

• One technique: multiply numerator and denominator by what is known as the conjugate.
• This enables removal of square roots.

Long Division

• Long division may be used to make canceling of factors easier.

Long Division Steps

• Divide: Divide the first term of the dividend by the first term of the divisor.
• Multiply: Multiply what you obtained by the divisor
• Subtract: Subtract and bring down the next term.

Explore the concept of indeterminate quantities in mathematics, focusing on the division of zero by zero. Learn about the difference between definite and indeterminate quantities. Understand why 0/0 is indeterminate due to multiple possible solutions.