Asymptotes and Discontinuities in Functions

The video tutorial explains how to find horizontal and vertical asymptotes for rational functions through six examples. It covers the rules for identifying vertical asymptotes by setting the denominator to zero and horizontal asymptotes by comparing the degrees of the numerator and denominator. Each example demonstrates different scenarios, including removable discontinuities and cases with no vertical asymptotes.

What is the vertical asymptote of the function f(x) = 2x / (x - 3)?

• x = 2

• x = -3

• x = 3

• x = 0

For the function f(x) = 2x / (x - 3), what is the horizontal asymptote?

• y = 3

• y = 0

• y = 1

• y = 2

In the function g(x) = (x - 4) / x^2, what is the vertical asymptote?

• x = 4

• x = 0

• x = 2

• x = -4

What is the horizontal asymptote for the function g(x) = (x - 4) / x^2?

• y = 1

• y = 0

• y = 2

• y = -1

For the function h(x) = (x^2 + 4) / (4 - x^2), what are the vertical asymptotes?

• x = 2 and x = -2

• x = 3 and x = -3

• x = 1 and x = -1

• x = 0 and x = 4

What is the horizontal asymptote of the function h(x) = (x^2 + 4) / (4 - x^2)?

• y = 2

• y = 1

• y = 0

• y = -1

In the function f(x) = (x^3 + 2) / (x - 1), why is there no horizontal asymptote?

• The degree of the numerator is less than the denominator.

• The degree of the numerator is equal to the denominator.

• The degree of the numerator is greater than the denominator.

• The function is not a rational function.

For the function g(x) = 5 / (x^2 + 1), what is the horizontal asymptote?

• y = 0

• y = 1

• y = 5

• y = -1

Why does the function h(x) = (x - 1) / (x^2 + 4x - 5) have a removable discontinuity?

• The function has no vertical asymptotes.

• The function is not defined for any x.

• The numerator and denominator have a common factor.

• The degree of the numerator is greater than the denominator.

What is the vertical asymptote for the function h(x) = (x - 1) / (x^2 + 4x - 5) after removing the discontinuity?

• x = 1

• x = -5

• x = 5

• x = 0