Limits at Infinity and Horizontal Asymptotes Explained

Mr. Bean's lesson covers horizontal asymptotes and their relation to limits, focusing on graph behavior and the rules for determining limits. The video explains how to analyze growth rates of functions to find asymptotes, provides examples, and discusses special cases involving infinity. It also covers trigonometric limits and concludes with test preparation advice.

What does a horizontal asymptote represent in the context of a graph?

• The maximum value of the graph

• The end behavior of a graph as x approaches infinity or negative infinity

• The point where the graph intersects the x-axis

• The point where the graph intersects the y-axis

Which of the following is true about horizontal asymptotes?

• A graph can cross a horizontal asymptote only at the origin

• A graph can only cross a horizontal asymptote once

• A graph can cross a horizontal asymptote an infinite number of times

• A graph can never cross a horizontal asymptote

If the denominator of a function grows faster than the numerator, what is the horizontal asymptote?

• y = -1

• y = infinity

• y = 0

• y = 1

What happens if the numerator and denominator of a function grow at the same rate?

• The horizontal asymptote is determined by the coefficients of the highest degree terms

• The horizontal asymptote is y = 1

• There is no horizontal asymptote

• The horizontal asymptote is y = 0

Which type of function grows the fastest as x approaches infinity?

• Linear functions

• Logarithmic functions

• Polynomial functions

• Exponential functions

For the function f(x) = (3x^2 + 2)/(x^2 + 1), what is the horizontal asymptote?

• y = 0

• y = 3/1

• y = 3

• y = 1

If the numerator of a function grows faster than the denominator, what is the horizontal asymptote?

• There is no horizontal asymptote

• y = 1

• y = infinity

• y = 0

What is the horizontal asymptote for the function f(x) = (2x)/(3x^2)?

• y = 1

• y = infinity

• y = 0

• y = 2/3

What is the limit of sin(x)/x as x approaches infinity?

• 1

• 0

• Infinity

• Does not exist

What happens to the function 5x * cos(x) as x approaches infinity?

• It approaches zero

• It approaches infinity

• It oscillates and does not approach a specific value

• It approaches a specific value