Glossary

Jump Discontinuity

Criticality: 3

A type of discontinuity where the function's value abruptly changes at a specific point, causing the left-hand limit and the right-hand limit to be different, and thus the overall limit does not exist.

Example:

A common example of a jump discontinuity is a step function, where the graph literally 'jumps' from one y-value to another at a specific x-value, making the overall limit at that point undefined.

Left-Hand Limit

Criticality: 3

The value a function approaches as its input approaches a specific point from values less than that point (from the left side).

Example:

If a piecewise function jumps at x=0, the left-hand limit as x approaches 0 might be 2, meaning the function approaches y=2 when x is slightly less than 0.

Limit

Criticality: 3

The value a function approaches as its input (x-value) approaches a specific point, representing the trend of the function rather than its exact value at that point.

Example:

For the function f(x) = x^2, the limit as x approaches 2 is 4, even if there was a hole at (2,4).

Oscillating Function

Criticality: 2

A function that fluctuates rapidly between different y-values as the input approaches a certain point, preventing it from settling on a single limit value.

Example:

The function f(x) = sin(1/x) is an oscillating function near x=0, as it wiggles infinitely many times between -1 and 1, so its limit at x=0 does not exist.

Right-Hand Limit

Criticality: 3

The value a function approaches as its input approaches a specific point from values greater than that point (from the right side).

Example:

For the same piecewise function, the right-hand limit as x approaches 0 might be 5, indicating the function approaches y=5 when x is slightly greater than 0.

Unbounded Function

Criticality: 2

A function whose values increase or decrease without bound (approach infinity or negative infinity) as the input approaches a certain point, causing the limit to not exist.

Example:

The function f(x) = 1/x^2 is an unbounded function as x approaches 0, because its y-values shoot up to positive infinity.

Vertical Asymptote

Criticality: 2

A vertical line that the graph of a function approaches but never touches, typically occurring where the function's value becomes unbounded.

Example:

For the function f(x) = 1/(x-3), there is a vertical asymptote at x=3, indicating the function's values go to infinity or negative infinity as x gets closer to 3.

Glossary

Jump Discontinuity

Criticality: 3

Example:

A common example of a jump discontinuity is a step function, where the graph literally 'jumps' from one y-value to another at a specific x-value, making the overall limit at that point undefined.

Left-Hand Limit

Criticality: 3

The value a function approaches as its input approaches a specific point from values less than that point (from the left side).

Example:

If a piecewise function jumps at x=0, the left-hand limit as x approaches 0 might be 2, meaning the function approaches y=2 when x is slightly less than 0.

Limit

Criticality: 3

The value a function approaches as its input (x-value) approaches a specific point, representing the trend of the function rather than its exact value at that point.

Example:

For the function f(x) = x^2, the limit as x approaches 2 is 4, even if there was a hole at (2,4).

Oscillating Function

Criticality: 2

A function that fluctuates rapidly between different y-values as the input approaches a certain point, preventing it from settling on a single limit value.

Example:

The function f(x) = sin(1/x) is an oscillating function near x=0, as it wiggles infinitely many times between -1 and 1, so its limit at x=0 does not exist.

Right-Hand Limit

Criticality: 3

The value a function approaches as its input approaches a specific point from values greater than that point (from the right side).

Example:

For the same piecewise function, the right-hand limit as x approaches 0 might be 5, indicating the function approaches y=5 when x is slightly greater than 0.

Unbounded Function

Criticality: 2

A function whose values increase or decrease without bound (approach infinity or negative infinity) as the input approaches a certain point, causing the limit to not exist.

Example:

The function f(x) = 1/x^2 is an unbounded function as x approaches 0, because its y-values shoot up to positive infinity.

Vertical Asymptote

Criticality: 2

A vertical line that the graph of a function approaches but never touches, typically occurring where the function's value becomes unbounded.

Example:

For the function f(x) = 1/(x-3), there is a vertical asymptote at x=3, indicating the function's values go to infinity or negative infinity as x gets closer to 3.