Limits at Infinity

Front

What is the definition of a limit at infinity?

Back

A limit at infinity refers to the value that a function approaches as the input approaches infinity or negative infinity.

Front

How do you determine the limit of a rational function as x approaches infinity?

Back

To determine the limit of a rational function as x approaches infinity, compare the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the limit is 0. If they are equal, the limit is the ratio of the leading coefficients. If the numerator's degree is greater, the limit is ±∞.

Front

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

Back

The limit of a constant function as x approaches infinity is the constant itself.

Front

What happens to the limit of a function if the degree of the numerator is greater than the degree of the denominator?

Back

If the degree of the numerator is greater than the degree of the denominator, the limit as x approaches infinity is ±∞, depending on the leading coefficients.

Front

What is the limit of the function f(x) = 1/x as x approaches infinity?

Back

The limit of f(x) = 1/x as x approaches infinity is 0.

Front

What is the limit of the function f(x) = x^2/(x^2 + 1) as x approaches infinity?

Back

The limit of f(x) = x^2/(x^2 + 1) as x approaches infinity is 1.

Front

How do horizontal asymptotes relate to limits at infinity?

Back

Horizontal asymptotes represent the value that a function approaches as x approaches infinity or negative infinity, indicating the limit at infinity.

Front

What is the limit of the function f(x) = 3x/(2x + 1) as x approaches infinity?

Back

The limit of f(x) = 3x/(2x + 1) as x approaches infinity is 3/2.

Front

What is the limit of the function f(x) = (2x^3 + 3)/(x^3 - 4) as x approaches infinity?

Back

The limit of f(x) = (2x^3 + 3)/(x^3 - 4) as x approaches infinity is 2.

Front

What is the limit of the function f(x) = (x^2 - 1)/(x^2 + 1) as x approaches infinity?

Back

The limit of f(x) = (x^2 - 1)/(x^2 + 1) as x approaches infinity is 1.