Limits and Continuity Review Flashcard--Ms.Li

Front

What is the definition of a limit in calculus?

Back

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

Front

What does it mean for a function to be continuous at a point?

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 removable discontinuity?

Back

A removable discontinuity occurs when a function is not defined at a point, but the limit exists at that point.

Front

What is a jump discontinuity?

Back

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

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.

Front

What is the formal definition of a limit?

Back

The formal definition states that \( \lim_{x \to a} f(x) = L \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).

Front

What is the limit of \( f(x) = \frac{x^2 - 4}{x - 2} \) as \( x \to 2 \)?

Back

The limit is 4.

Front

What must be true if \( \lim_{x \to a} f(x) \) exists?

Back

The left and right limits as \( x \to a \) must be equal.

Front

What is the limit of \( \frac{x^3 - x^2 + 1}{5 - 3x - x^4} \) as \( x \to \infty \)?

Back

The limit is 0.

Front

Which of the following is NOT a type of discontinuity?

Back

Rational is not a type of discontinuity.