Limits, Continuity, IVT

Front

What is a finite jump discontinuity?

Back

A finite jump discontinuity occurs when the left-hand limit and right-hand limit of a function at a point exist but are not equal, resulting in a 'jump' in the graph.

Tags

CCSS.8.F.A.3

Front

What is an infinite discontinuity?

Back

An infinite discontinuity occurs when the function approaches infinity or negative infinity as the input approaches a certain value, leading to a vertical asymptote.

Tags

CCSS.HSF-IF.C.7D

Front

What is a removable discontinuity?

Back

A removable discontinuity occurs when a function is not defined at a point, but the limit exists at that point. It can be 'removed' by redefining the function at that point.

Tags

CCSS.HSF-IF.C.7D

Front

State the Intermediate Value Theorem (IVT).

Back

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], and N is any number between f(a) and f(b), then there exists at least one c in (a, b) such that f(c) = N.

Front

How do you determine if a function is continuous at a point?

Back

A function f is continuous at a point a if: 1) f(a) is defined, 2) the limit of f(x) as x approaches a exists, and 3) the limit equals f(a).

Tags

CCSS.HSF.IF.A.2

Front

What does it mean for a function to be continuous on an interval?

Back

A function is continuous on an interval if it is continuous at every point within that interval.

Front

Provide an example of a function with a finite jump discontinuity.

Back

The piecewise function f(x) = { 1 for x < 0, 2 for x = 0, 3 for x > 0 } has a finite jump discontinuity at x = 0.

Tags

CCSS.HSF-IF.C.7B

Front

Provide an example of a function with an infinite discontinuity.

Back

The function f(x) = 1/(x-1) has an infinite discontinuity at x = 1.

Tags

CCSS.HSF-IF.C.7D

Front

What is the significance of the limits in determining continuity?

Back

Limits help determine continuity by showing whether the function approaches a specific value as the input approaches a point.

Front

How can you identify intervals where a polynomial function is positive?

Back

To find intervals where a polynomial function is positive, identify the roots and test the sign of the function in the intervals defined by these roots.