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Glossary

C

Constant Limit

Criticality: 1

The limit of a constant function is simply the constant itself, as the function's value does not change regardless of what 'x' approaches.

Example:

The constant limit of limx→107lim_{x \to 10} 7 is simply 7, because the function is always 7, no matter what x approaches.

Constant Multiple Rule

Criticality: 2

States that the limit of a constant times a function is the constant times the limit of the function.

Example:

To evaluate limx→2(5x3)lim_{x \to 2} (5x^3), the Constant Multiple Rule allows you to pull out the 5: 5⋅limx→2x3=5⋅8=405 \cdot lim_{x \to 2} x^3 = 5 \cdot 8 = 40.

Continuity

Criticality: 3

A property of a function where its graph can be drawn without lifting the pen, meaning the limit exists, the function is defined at that point, and the limit equals the function's value.

Example:

A polynomial function like f(x)=x2+3f(x) = x^2 + 3 exhibits continuity everywhere, as its graph has no breaks, jumps, or holes.

D

Difference Rule

Criticality: 2

States that the limit of a difference of two functions is the difference of their individual limits.

Example:

To find limx→4(x2−x)lim_{x \to 4} (x^2 - x), you can apply the Difference Rule to evaluate limx→4x2−limx→4x=16−4=12lim_{x \to 4} x^2 - lim_{x \to 4} x = 16 - 4 = 12.

Direct Substitution

Criticality: 3

The primary method for evaluating a limit, where the value that 'x' approaches is directly plugged into the function.

Example:

When evaluating limx→5(2x−1)lim_{x \to 5} (2x - 1), the first step is direct substitution, yielding 2(5)−1=92(5) - 1 = 9.

I

Indeterminate Form

Criticality: 3

An expression, such as 0/0 or $\infty/\infty$, that arises when evaluating a limit by direct substitution and does not immediately provide the limit's value, requiring further algebraic manipulation or other techniques.

Example:

When evaluating limx→2x2−4x−2lim_{x \to 2} \frac{x^2 - 4}{x - 2}, direct substitution yields 00\frac{0}{0}, which is an indeterminate form, signaling the need for factoring.

L

L'Hôpital's Rule

Criticality: 2

A technique used to evaluate indeterminate forms of limits by taking the derivatives of the numerator and denominator separately.

Example:

For limx→0sin⁡xxlim_{x \to 0} \frac{\sin x}{x}, which is an indeterminate form 00\frac{0}{0}, L'Hôpital's Rule allows you to take derivatives to get limx→0cos⁡x1=1lim_{x \to 0} \frac{\cos x}{1} = 1.

Limit Laws

Criticality: 3

A set of rules that allow for the evaluation of limits of combined functions by breaking them down into simpler, individual limits.

Example:

Using the limit laws, you can evaluate limx→1(x2+3x)lim_{x \to 1} (x^2 + 3x) by finding the limit of x2x^2 and the limit of 3x3x separately, then adding the results.

Limits Algebraically

Criticality: 3

A method of evaluating limits by applying algebraic operations and properties to the function rather than relying on graphical analysis.

Example:

To find the limit algebraically of limx→2(x2+3x)lim_{x \to 2} (x^2 + 3x), you would directly substitute x=2 to get 22+3(2)=102^2 + 3(2) = 10.

P

Piecewise Function

Criticality: 2

A function defined by multiple sub-functions, each applicable over a certain interval of the domain.

Example:

To determine the limit of a piecewise function like f(x)=x2 for x<0,x+1 for x≥0f(x) = {x^2 \text{ for } x<0, x+1 \text{ for } x \ge 0} at x=0x=0, you must evaluate the limit from the left and right using the appropriate sub-functions.

Power Rule (for Limits)

Criticality: 2

States that the limit of a function raised to a positive integer power is the limit of the function raised to that power.

Example:

To find limx→2(x2+1)3lim_{x \to 2} (x^2 + 1)^3, the Power Rule allows you to first find limx→2(x2+1)=5lim_{x \to 2} (x^2 + 1) = 5, then cube the result: 53=1255^3 = 125.

Product Rule

Criticality: 2

States that the limit of a product of two functions is the product of their individual limits.

Example:

If you need to find limx→1(x2⋅(x+1))lim_{x \to 1} (x^2 \cdot (x+1)), the Product Rule lets you calculate limx→1x2⋅limx→1(x+1)=1⋅2=2lim_{x \to 1} x^2 \cdot lim_{x \to 1} (x+1) = 1 \cdot 2 = 2.

Q

Quotient Rule

Criticality: 2

States that the limit of a quotient of two functions is the quotient of their individual limits, provided the limit of the denominator is not zero.

Example:

To evaluate limx→3x+1x−1lim_{x \to 3} \frac{x+1}{x-1}, the Quotient Rule applies: limx→3(x+1)limx→3(x−1)=42=2\frac{lim_{x \to 3} (x+1)}{lim_{x \to 3} (x-1)} = \frac{4}{2} = 2.

R

Root Rule (for Limits)

Criticality: 2

States that the limit of the nth root of a function is the nth root of the limit of the function.

Example:

When evaluating limx→7x+2lim_{x \to 7} \sqrt{x+2}, the Root Rule means you can find limx→7(x+2)=9lim_{x \to 7} (x+2) = 9, then take the square root: 9=3\sqrt{9} = 3.

S

Sum Rule

Criticality: 2

States that the limit of a sum of two functions is the sum of their individual limits.

Example:

If limx→0f(x)=2lim_{x \to 0} f(x) = 2 and limx→0g(x)=3lim_{x \to 0} g(x) = 3, then by the Sum Rule, limx→0[f(x)+g(x)]=2+3=5lim_{x \to 0} [f(x) + g(x)] = 2 + 3 = 5.