Glossary

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 $lim_{x \to 10} 7$ 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 $lim_{x \to 2} (5x^3)$ , the Constant Multiple Rule allows you to pull out the 5: $5 \cdot lim_{x \to 2} x^3 = 5 \cdot 8 = 40$ .

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 $f(x) = x^2 + 3$ exhibits continuity everywhere, as its graph has no breaks, jumps, or holes.

Difference Rule

Criticality: 2

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

Example:

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

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 $lim_{x \to 5} (2x - 1)$ , the first step is direct substitution, yielding $2(5) - 1 = 9$ .

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 $lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ , direct substitution yields $\frac{0}{0}$ , which is an indeterminate form, signaling the need for factoring.

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 $lim_{x \to 0} \frac{\sin x}{x}$ , which is an indeterminate form $\frac{0}{0}$ , L'Hôpital's Rule allows you to take derivatives to get $lim_{x \to 0} \frac{\cos x}{1} = 1$ .

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 $lim_{x \to 1} (x^2 + 3x)$ by finding the limit of $x^2$ and the limit of $3x$ 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 $lim_{x \to 2} (x^2 + 3x)$ , you would directly substitute x=2 to get $2^2 + 3(2) = 10$ .

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 $f(x) = {x^2 \text{ for } x<0, x+1 \text{ for } x \ge 0}$ at $x=0$ , 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 $lim_{x \to 2} (x^2 + 1)^3$ , the Power Rule allows you to first find $lim_{x \to 2} (x^2 + 1) = 5$ , then cube the result: $5^3 = 125$ .

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 $lim_{x \to 1} (x^2 \cdot (x+1))$ , the Product Rule lets you calculate $lim_{x \to 1} x^2 \cdot lim_{x \to 1} (x+1) = 1 \cdot 2 = 2$ .

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 $lim_{x \to 3} \frac{x+1}{x-1}$ , the Quotient Rule applies: $\frac{lim_{x \to 3} (x+1)}{lim_{x \to 3} (x-1)} = \frac{4}{2} = 2$ .

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

Sum Rule

Criticality: 2

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

Example:

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

Glossary

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 $lim_{x \to 10} 7$ 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 $lim_{x \to 2} (5x^3)$ , the Constant Multiple Rule allows you to pull out the 5: $5 \cdot lim_{x \to 2} x^3 = 5 \cdot 8 = 40$ .

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 $f(x) = x^2 + 3$ exhibits continuity everywhere, as its graph has no breaks, jumps, or holes.

Difference Rule

Criticality: 2

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

Example:

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

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 $lim_{x \to 5} (2x - 1)$ , the first step is direct substitution, yielding $2(5) - 1 = 9$ .

Indeterminate Form

Criticality: 3

Example:

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

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 $lim_{x \to 0} \frac{\sin x}{x}$ , which is an indeterminate form $\frac{0}{0}$ , L'Hôpital's Rule allows you to take derivatives to get $lim_{x \to 0} \frac{\cos x}{1} = 1$ .

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 $lim_{x \to 1} (x^2 + 3x)$ by finding the limit of $x^2$ and the limit of $3x$ 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 $lim_{x \to 2} (x^2 + 3x)$ , you would directly substitute x=2 to get $2^2 + 3(2) = 10$ .

Piecewise Function

Criticality: 2

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

Example:

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

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 $lim_{x \to 1} (x^2 \cdot (x+1))$ , the Product Rule lets you calculate $lim_{x \to 1} x^2 \cdot lim_{x \to 1} (x+1) = 1 \cdot 2 = 2$ .

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 $lim_{x \to 3} \frac{x+1}{x-1}$ , the Quotient Rule applies: $\frac{lim_{x \to 3} (x+1)}{lim_{x \to 3} (x-1)} = \frac{4}{2} = 2$ .

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

Sum Rule

Criticality: 2

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

Example:

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