Glossary

Composite Functions (Limits)

Criticality: 2

For a limit of a composite function $f(g(x))$, one first evaluates the limit of the inner function $g(x)$ and then applies the outer function $f$ to that result.

Example:

To find $\lim_{x o 0} \sin(e^x)$ , we first find $\lim_{x o 0} e^x = 1$ , then evaluate $\sin(1)$ , demonstrating the limit of a composite function.

Conjugates (for limits)

Criticality: 2

An algebraic technique used to simplify expressions involving radicals in limits, typically by multiplying the numerator and denominator by the conjugate of the radical expression.

Example:

To evaluate $\lim_{x o 0} \frac{\sqrt{x+4}-2}{x}$ , multiply by the conjugate $\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}$ to simplify the expression.

Continuity

Criticality: 3

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

Example:

A polynomial function like $f(x) = x^3 - 2x + 1$ is continuous everywhere, meaning its limit at any point equals its function value at that point.

Determining Limits From A Graph

Criticality: 2

A method of finding a limit by observing the y-value a function's graph approaches as x gets closer to a specific value or infinity.

Example:

On the graph of $y = \frac{1}{x}$ , as x approaches positive infinity, the y-value approaches 0, indicating a graphical limit of 0.

Determining Limits Using Algebraic Manipulation

Criticality: 3

A set of techniques used to simplify a function, often to eliminate indeterminate forms, before evaluating its limit.

Example:

When finding $\lim_{x o 3} \frac{x^2-9}{x-3}$ , one must use algebraic manipulation (factoring) to simplify the expression to $x+3$ before plugging in $x=3$ .

Determining Limits Using Algebraic Properties

Criticality: 3

A set of rules (limit laws) that allow for the evaluation of limits of complex functions by breaking them down into simpler components, such as sums, differences, products, and quotients of individual limits.

Example:

Using algebraic properties of limits, $\lim_{x o 2} (x^2 + 3x)$ can be evaluated as $\lim_{x o 2} x^2 + \lim_{x o 2} 3x = 4 + 6 = 10$ .

Difference Limit Law

Criticality: 2

States that the limit of a difference of two functions is equal to the difference of their individual limits, provided each limit exists.

Example:

To find $\lim_{x o 5} (x^2 - \sqrt{x})$ , you can apply the Difference Limit Law to get $\lim_{x o 5} x^2 - \lim_{x o 5} \sqrt{x} = 25 - \sqrt{5}$ .

Estimating Limit Values From Tables

Criticality: 2

A method of approximating a limit by analyzing the trend of y-values in a table as x-values get progressively closer to a specific point from both sides.

Example:

If a table shows $f(x)$ values of 2.9, 2.99, 2.999 as x approaches 3 from the left, and 3.001, 3.01, 3.1 as x approaches 3 from the right, the tabular limit is estimated to be 3.

Indeterminate Forms

Criticality: 3

Expressions like 0/0 or $\infty/\infty$ that arise when directly substituting the limit value into a function, indicating that further algebraic manipulation or L'Hôpital's Rule is required to find the limit.

Example:

When evaluating $\lim_{x o 1} \frac{x-1}{\ln x}$ , direct substitution yields $\frac{0}{0}$ , which is an indeterminate form, signaling the need for L'Hôpital's Rule.

L'Hôpital's Rule

Criticality: 3

A rule used to evaluate limits of indeterminate forms (0/0 or $\infty/\infty$) by taking the derivatives of the numerator and denominator separately.

Example:

To find $\lim_{x o 0} \frac{\sin x}{x}$ , since it's an indeterminate form $\frac{0}{0}$ , apply L'Hôpital's Rule to get $\lim_{x o 0} \frac{\cos x}{1} = 1$ .

Limit

Criticality: 3

The value that a function approaches as the input (x) approaches a certain value. It describes the behavior of a function near a point, not necessarily at the point itself.

Example:

For $f(x) = x^2$ , the limit as x approaches 2 is 4, because as x gets closer to 2, $x^2$ gets closer to 4.

Piecewise Functions (Limits)

Criticality: 2

Functions defined by multiple sub-functions, each applicable over a certain interval, requiring evaluation of one-sided limits at the boundary points to determine the overall limit or continuity.

Example:

To find the limit of a piecewise function at $x=0$ where $f(x) = x^2$ for $x<0$ and $f(x) = x+1$ for $x \ge 0$ , you must check the left-hand limit ( $\lim_{x o 0^-} x^2 = 0$ ) and the right-hand limit ( $\lim_{x o 0^+} (x+1) = 1$ ).

Product Limit Law

Criticality: 2

States that the limit of a product of two functions is equal to the product of their individual limits, provided each limit exists.

Example:

The Product Limit Law allows us to evaluate $\lim_{x o 1} (x^3 \cdot \cos(x))$ as $(\lim_{x o 1} x^3) \cdot (\lim_{x o 1} \cos(x)) = 1 \cdot \cos(1)$ .

Quotient Limit Law

Criticality: 2

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

Example:

Using the Quotient Limit Law, $\lim_{x o 2} \frac{x+1}{x-1}$ becomes $\frac{\lim_{x o 2} (x+1)}{\lim_{x o 2} (x-1)} = \frac{3}{1} = 3$ .

Simplifying Rational Functions (for limits)

Criticality: 2

An algebraic technique for evaluating limits of rational functions by factoring the numerator and denominator to cancel common terms, especially when direct substitution results in an indeterminate form.

Example:

For $\lim_{x o -2} \frac{x^2-4}{x+2}$ , simplifying rational functions by factoring the numerator to $(x-2)(x+2)$ allows cancellation of $(x+2)$ , leading to $\lim_{x o -2} (x-2) = -4$ .

Squeeze Theorem

Criticality: 2

A theorem stating that if a function is 'squeezed' between two other functions that have the same limit at a certain point, then the squeezed function also has that same limit at that point.

Example:

If we know that $-x^2 \le x^2 \sin(\frac{1}{x}) \le x^2$ for $x eq 0$ , and $\lim_{x o 0} (-x^2) = 0$ and $\lim_{x o 0} (x^2) = 0$ , then by the Squeeze Theorem, $\lim_{x o 0} x^2 \sin(\frac{1}{x}) = 0$ .

Sum Limit Law

Criticality: 2

States that the limit of a sum of two functions is equal to the sum of their individual limits, provided each limit exists.

Example:

If $\lim_{x o c} f(x) = L$ and $\lim_{x o c} g(x) = M$ , then $\lim_{x o c} [f(x) + g(x)] = L + M$ by the Sum Limit Law.

Glossary

Composite Functions (Limits)

Criticality: 2

For a limit of a composite function $f(g(x))$, one first evaluates the limit of the inner function $g(x)$ and then applies the outer function $f$ to that result.

Example:

To find $\lim_{x o 0} \sin(e^x)$ , we first find $\lim_{x o 0} e^x = 1$ , then evaluate $\sin(1)$ , demonstrating the limit of a composite function.

Conjugates (for limits)

Criticality: 2

An algebraic technique used to simplify expressions involving radicals in limits, typically by multiplying the numerator and denominator by the conjugate of the radical expression.

Example:

To evaluate $\lim_{x o 0} \frac{\sqrt{x+4}-2}{x}$ , multiply by the conjugate $\frac{\sqrt{x+4}+2}{\sqrt{x+4}+2}$ to simplify the expression.

Continuity

Criticality: 3

Example:

A polynomial function like $f(x) = x^3 - 2x + 1$ is continuous everywhere, meaning its limit at any point equals its function value at that point.

Determining Limits From A Graph

Criticality: 2

A method of finding a limit by observing the y-value a function's graph approaches as x gets closer to a specific value or infinity.

Example:

On the graph of $y = \frac{1}{x}$ , as x approaches positive infinity, the y-value approaches 0, indicating a graphical limit of 0.

Determining Limits Using Algebraic Manipulation

Criticality: 3

A set of techniques used to simplify a function, often to eliminate indeterminate forms, before evaluating its limit.

Example:

When finding $\lim_{x o 3} \frac{x^2-9}{x-3}$ , one must use algebraic manipulation (factoring) to simplify the expression to $x+3$ before plugging in $x=3$ .

Determining Limits Using Algebraic Properties

Criticality: 3

Example:

Using algebraic properties of limits, $\lim_{x o 2} (x^2 + 3x)$ can be evaluated as $\lim_{x o 2} x^2 + \lim_{x o 2} 3x = 4 + 6 = 10$ .

Difference Limit Law

Criticality: 2

States that the limit of a difference of two functions is equal to the difference of their individual limits, provided each limit exists.

Example:

To find $\lim_{x o 5} (x^2 - \sqrt{x})$ , you can apply the Difference Limit Law to get $\lim_{x o 5} x^2 - \lim_{x o 5} \sqrt{x} = 25 - \sqrt{5}$ .

Estimating Limit Values From Tables

Criticality: 2

A method of approximating a limit by analyzing the trend of y-values in a table as x-values get progressively closer to a specific point from both sides.

Example:

If a table shows $f(x)$ values of 2.9, 2.99, 2.999 as x approaches 3 from the left, and 3.001, 3.01, 3.1 as x approaches 3 from the right, the tabular limit is estimated to be 3.

Indeterminate Forms

Criticality: 3

Example:

When evaluating $\lim_{x o 1} \frac{x-1}{\ln x}$ , direct substitution yields $\frac{0}{0}$ , which is an indeterminate form, signaling the need for L'Hôpital's Rule.

L'Hôpital's Rule

Criticality: 3

A rule used to evaluate limits of indeterminate forms (0/0 or $\infty/\infty$) by taking the derivatives of the numerator and denominator separately.

Example:

To find $\lim_{x o 0} \frac{\sin x}{x}$ , since it's an indeterminate form $\frac{0}{0}$ , apply L'Hôpital's Rule to get $\lim_{x o 0} \frac{\cos x}{1} = 1$ .

Limit

Criticality: 3

The value that a function approaches as the input (x) approaches a certain value. It describes the behavior of a function near a point, not necessarily at the point itself.

Example:

For $f(x) = x^2$ , the limit as x approaches 2 is 4, because as x gets closer to 2, $x^2$ gets closer to 4.

Piecewise Functions (Limits)

Criticality: 2

Functions defined by multiple sub-functions, each applicable over a certain interval, requiring evaluation of one-sided limits at the boundary points to determine the overall limit or continuity.

Example:

Product Limit Law

Criticality: 2

States that the limit of a product of two functions is equal to the product of their individual limits, provided each limit exists.

Example:

The Product Limit Law allows us to evaluate $\lim_{x o 1} (x^3 \cdot \cos(x))$ as $(\lim_{x o 1} x^3) \cdot (\lim_{x o 1} \cos(x)) = 1 \cdot \cos(1)$ .

Quotient Limit Law

Criticality: 2

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

Example:

Using the Quotient Limit Law, $\lim_{x o 2} \frac{x+1}{x-1}$ becomes $\frac{\lim_{x o 2} (x+1)}{\lim_{x o 2} (x-1)} = \frac{3}{1} = 3$ .

Simplifying Rational Functions (for limits)

Criticality: 2

Example:

For $\lim_{x o -2} \frac{x^2-4}{x+2}$ , simplifying rational functions by factoring the numerator to $(x-2)(x+2)$ allows cancellation of $(x+2)$ , leading to $\lim_{x o -2} (x-2) = -4$ .

Squeeze Theorem

Criticality: 2

A theorem stating that if a function is 'squeezed' between two other functions that have the same limit at a certain point, then the squeezed function also has that same limit at that point.

Example:

Sum Limit Law

Criticality: 2

States that the limit of a sum of two functions is equal to the sum of their individual limits, provided each limit exists.

Example:

If $\lim_{x o c} f(x) = L$ and $\lim_{x o c} g(x) = M$ , then $\lim_{x o c} [f(x) + g(x)] = L + M$ by the Sum Limit Law.