Glossary

Algebraic Representation

Criticality: 3

Determining the exact value of a limit by manipulating the function's equation using algebraic techniques like factoring, rationalizing, or applying limit properties.

Example:

Using algebraic representation, we can find $\lim_{x \to 2} \frac{x^2-4}{x-2}$ by factoring the numerator to $(x-2)(x+2)$ , canceling $(x-2)$ , and then substituting x=2 into $(x+2)$ to get 4.

Connecting Multiple Representations of Limits

Criticality: 3

The ability to understand and interpret limits using graphs, tables of values, and algebraic equations, and to switch between these representations.

Example:

To confirm $\lim_{x \to 2} \frac{x^2-4}{x-2} = 4$ , you can simplify the expression algebraically, observe the function's behavior on a graph, or check values in a table of values.

Continuity

Criticality: 3

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

Example:

A polynomial function like $f(x) = x^3 - 2x + 5$ is continuous everywhere, as there are no breaks, jumps, or holes in its graph.

Differentiability

Criticality: 3

A property of a function at a point where its derivative exists, implying the function is smooth and has no sharp corners, cusps, or vertical tangents at that point.

Example:

The absolute value function $f(x) = |x|$ is continuous at x=0 but not differentiable at x=0 because it has a sharp corner there.

Graphical Representation

Criticality: 2

Approximating a limit by visually inspecting the behavior of a function's graph as x approaches a specific value from both the left and the right.

Example:

On a graphical representation, if the curve approaches a y-value of 3 as x approaches 1 from both sides, then the limit at x=1 is 3, even if there's a hole at (1,3).

Left-hand Limit

Criticality: 2

The value that a function approaches as the input (x) approaches a certain value from values less than it (from the left side on a graph).

Example:

For a piecewise function, if $f(x) = x+1$ for $x<2$ , the left-hand limit as x approaches 2 is $2+1=3$ .

Limit

Criticality: 3

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

Example:

For $f(x) = x+2$ , the limit as x approaches 3 is 5, because as x gets closer to 3, f(x) gets closer to 5.

Numerical Representation (Tables)

Criticality: 2

Approximating a limit by observing the trend of function values (f(x)) as input values (x) get progressively closer to a target value from both sides.

Example:

A numerical representation for $\lim_{x \to 0} \frac{\sin x}{x}$ would show f(x) values like 0.998, 0.999, 1.000, 0.999, 0.998 as x approaches 0, suggesting the limit is 1.

Removable Discontinuity

Criticality: 2

A type of discontinuity in a function where the limit exists at a point, but the function value at that point is either undefined or different from the limit. It appears as a 'hole' in the graph.

Example:

The function $f(x) = \frac{x^2-1}{x-1}$ has a removable discontinuity at x=1 because the limit as x approaches 1 is 2, but f(1) is undefined.

Right-hand Limit

Criticality: 2

The value that a function approaches as the input (x) approaches a certain value from values greater than it (from the right side on a graph).

Example:

For a piecewise function, if $f(x) = x^2$ for $x \ge 2$ , the right-hand limit as x approaches 2 is $2^2=4$ .

Glossary

Algebraic Representation

Criticality: 3

Determining the exact value of a limit by manipulating the function's equation using algebraic techniques like factoring, rationalizing, or applying limit properties.

Example:

Using algebraic representation, we can find $\lim_{x \to 2} \frac{x^2-4}{x-2}$ by factoring the numerator to $(x-2)(x+2)$ , canceling $(x-2)$ , and then substituting x=2 into $(x+2)$ to get 4.

Connecting Multiple Representations of Limits

Criticality: 3

The ability to understand and interpret limits using graphs, tables of values, and algebraic equations, and to switch between these representations.

Example:

To confirm $\lim_{x \to 2} \frac{x^2-4}{x-2} = 4$ , you can simplify the expression algebraically, observe the function's behavior on a graph, or check values in a table of values.

Continuity

Criticality: 3

Example:

A polynomial function like $f(x) = x^3 - 2x + 5$ is continuous everywhere, as there are no breaks, jumps, or holes in its graph.

Differentiability

Criticality: 3

A property of a function at a point where its derivative exists, implying the function is smooth and has no sharp corners, cusps, or vertical tangents at that point.

Example:

The absolute value function $f(x) = |x|$ is continuous at x=0 but not differentiable at x=0 because it has a sharp corner there.

Graphical Representation

Criticality: 2

Approximating a limit by visually inspecting the behavior of a function's graph as x approaches a specific value from both the left and the right.

Example:

On a graphical representation, if the curve approaches a y-value of 3 as x approaches 1 from both sides, then the limit at x=1 is 3, even if there's a hole at (1,3).

Left-hand Limit

Criticality: 2

The value that a function approaches as the input (x) approaches a certain value from values less than it (from the left side on a graph).

Example:

For a piecewise function, if $f(x) = x+1$ for $x<2$ , the left-hand limit as x approaches 2 is $2+1=3$ .

Limit

Criticality: 3

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

Example:

For $f(x) = x+2$ , the limit as x approaches 3 is 5, because as x gets closer to 3, f(x) gets closer to 5.

Numerical Representation (Tables)

Criticality: 2

Approximating a limit by observing the trend of function values (f(x)) as input values (x) get progressively closer to a target value from both sides.

Example:

A numerical representation for $\lim_{x \to 0} \frac{\sin x}{x}$ would show f(x) values like 0.998, 0.999, 1.000, 0.999, 0.998 as x approaches 0, suggesting the limit is 1.

Removable Discontinuity

Criticality: 2

A type of discontinuity in a function where the limit exists at a point, but the function value at that point is either undefined or different from the limit. It appears as a 'hole' in the graph.

Example:

The function $f(x) = \frac{x^2-1}{x-1}$ has a removable discontinuity at x=1 because the limit as x approaches 1 is 2, but f(1) is undefined.

Right-hand Limit

Criticality: 2

The value that a function approaches as the input (x) approaches a certain value from values greater than it (from the right side on a graph).

Example:

For a piecewise function, if $f(x) = x^2$ for $x \ge 2$ , the right-hand limit as x approaches 2 is $2^2=4$ .