Continuity & Limits

Front

What is continuity in a function?

Back

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.

Front

What is a limit in calculus?

Back

A limit is the value that a function approaches as the input approaches a certain point.

Front

What does it mean for a function to have an infinite discontinuity?

Back

An infinite discontinuity occurs when the function approaches infinity or negative infinity as the input approaches a certain value.

Front

What is a jump discontinuity?

Back

A jump discontinuity occurs when the left-hand limit and the right-hand limit at a certain point exist but are not equal.

Front

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

Back

A function is continuous at a point if: 1) the function is defined at that point, 2) the limit exists at that point, and 3) the limit equals the function's value at that point.

Front

What is the formal definition of a limit?

Back

The formal definition of a limit states that for every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Front

What is the difference between left-hand limit and right-hand limit?

Back

The left-hand limit is the value that f(x) approaches as x approaches a from the left, while the right-hand limit is the value that f(x) approaches as x approaches a from the right.

Front

What is the significance of the limit notation lim x→a f(x)?

Back

This notation indicates the value that the function f(x) approaches as x gets arbitrarily close to a.

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, allowing the function to be redefined to make it continuous.

Front

What is the Intermediate Value Theorem?

Back

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b) at least once.