Limits with Infinity

Front

What is the limit of a function as x approaches infinity?

Back

The limit of a function as x approaches infinity describes the behavior of the function as the input grows larger and larger. It can approach a finite number, infinity, or negative infinity.

Front

Define continuity at a point.

Back

A function is continuous at a point x = c if the following three conditions are met: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit equals f(c).

Front

Evaluate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4)

Back

The limit is 3/5, as the dominant terms in the numerator and denominator are both x^2.

Front

What is a removable point discontinuity?

Back

A removable point discontinuity occurs when a function is not defined at a point, but can be made continuous by defining or redefining the function at that point.

Front

Evaluate the limit: lim (x -> 0) (sin x)/x

Back

The limit is 1.

Front

What is a non-removable infinite discontinuity?

Back

A non-removable infinite discontinuity occurs when the function approaches infinity or negative infinity as x approaches a certain value.

Front

Evaluate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)

Back

The limit is 2, after factoring the numerator.

Front

What is the dominant term in a polynomial function?

Back

The dominant term is the term with the highest degree in the polynomial, which determines the behavior of the function as x approaches infinity.

Front

Evaluate the limit: lim (x -> ∞) (ln x)/x

Back

The limit is 0, as ln x grows slower than x.

Front

What is a non-removable jump discontinuity?

Back

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