Limits and Continuity Practice

Front

What does it mean for a function to be continuous on its domain?

Back

A function is continuous on its domain if there are no breaks, jumps, or holes in the graph of the function for all points in that domain.

Front

Is the function continuous on the interval \([-2, 3]\)?

Back

Yes, if the function does not have any discontinuities (like jumps or holes) within the interval \([-2, 3]\).

Front

What is a Jump Discontinuity?

Back

A Jump Discontinuity occurs when the left-hand limit and the right-hand limit at a point do not equal each other, causing a 'jump' in the graph.

Front

What is the condition for continuity at a point \(x = a\)?

Back

A function is continuous at \(x = a\) if: 1) \(f(a)\) is defined, 2) \(\lim_{x \to a} f(x)\) exists, and 3) \(\lim_{x \to a} f(x) = f(a)\).

Front

What does \(\lim_{x \to a^-} f(x)\) represent?

Back

It represents the limit of the function \(f(x)\) as \(x\) approaches \(a\) from the left side.

Front

What does \(\lim_{x \to a^+} f(x)\) represent?

Back

It represents the limit of the function \(f(x)\) as \(x\) approaches \(a\) from the right side.

Front

What is the significance of \(\lim_{x \to a} f(x) = f(a)\)?

Back

This condition indicates that the function is continuous at the point \(x = a\).

Front

What is an Infinite Discontinuity?

Back

An Infinite Discontinuity occurs when the function approaches infinity (or negative infinity) as \(x\) approaches a certain value.

Front

What is a Removable Discontinuity?

Back

A Removable Discontinuity occurs when a function has a hole at a point, but can be made continuous by defining or redefining the function at that point.

Front

How do you determine if a function is continuous on a closed interval?

Back

A function is continuous on a closed interval if it is continuous at every point in the interval and the endpoints are included.