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Glossary

A

Algebraic Manipulation for Limits

Criticality: 3

Techniques used to transform a limit expression into a simpler form, often to resolve indeterminate forms and allow for direct substitution.

Example:

When evaluating lim⁡xightarrow2x2−4x−2\lim_{x ightarrow 2} \frac{x^2 - 4}{x - 2}, we use algebraic manipulation to simplify the expression to lim⁡xightarrow2(x+2)\lim_{x ightarrow 2} (x+2) before finding the limit.

D

Denominator

Criticality: 1

The bottom part of a fraction, representing the divisor.

Example:

In the rational expression x2−5x−6x2−7x+6\frac{x^2 - 5x - 6}{x^2 - 7x + 6}, the polynomial x2−7x+6x^2 - 7x + 6 is the denominator.

F

Factoring

Criticality: 3

The process of breaking down an algebraic expression into a product of simpler expressions (factors).

Example:

To evaluate lim⁡xightarrow1x2−1x−1\lim_{x ightarrow 1} \frac{x^2 - 1}{x - 1}, we use factoring to rewrite the numerator as (x−1)(x+1)(x-1)(x+1), which allows for cancellation.

G

Greatest Common Factor (GCF)

Criticality: 2

The largest factor that two or more terms or numbers share.

Example:

When simplifying the expression 3x2+6x3x^2 + 6x, the greatest common factor is 3x3x, allowing us to factor it as 3x(x+2)3x(x+2).

L

Limit

Criticality: 3

The value that a function approaches as the input approaches a certain value.

Example:

The limit of f(x)=x2f(x) = x^2 as xx approaches 2 is 4, meaning f(x)f(x) gets arbitrarily close to 4 as xx gets close to 2.

Limit Does Not Exist

Criticality: 3

A situation where a function does not approach a single, finite value as the input approaches a certain point, often due to oscillation, unbounded behavior, or different values from the left and right.

Example:

For lim⁡xightarrow01x\lim_{x ightarrow 0} \frac{1}{x}, the limit does not exist because the function approaches positive infinity from the right and negative infinity from the left.

N

Numerator

Criticality: 1

The top part of a fraction, representing the dividend.

Example:

In the rational expression x2−5x−6x2−7x+6\frac{x^2 - 5x - 6}{x^2 - 7x + 6}, the polynomial x2−5x−6x^2 - 5x - 6 is the numerator.

R

Rational Functions

Criticality: 2

A function that can be expressed as the ratio of two polynomial functions, where the denominator is not zero.

Example:

The function f(x)=x2+3xx−5f(x) = \frac{x^2 + 3x}{x - 5} is a rational function often encountered when evaluating limits that require algebraic simplification.