Understanding Limits at Infinity with Exponential Functions

The video tutorial explores limits at infinity involving the exponential function with base e. It first examines the limit as x approaches infinity, showing that the function approaches zero. The tutorial then considers the limit as x approaches negative infinity, demonstrating that the function approaches -1. The video concludes with a graphical representation of the function, highlighting two horizontal asymptotes at y = 0 and y = -1, verifying the calculated limits.

What happens to e^x as x approaches infinity?

• It approaches zero.

• It remains constant.

• It approaches negative infinity.

• It approaches positive infinity.

Why does a fraction with a fixed numerator and an increasing denominator approach zero?

• Because the denominator decreases.

• Because the fraction becomes larger.

• Because the numerator also increases.

• Because the fraction becomes smaller.

In the second example, what is the limit as x approaches negative infinity?

• The limit is 1.

• The limit is -1.

• The limit is 3.

• The limit is 0.

How can e^x be rewritten to better understand its behavior as x approaches negative infinity?

• As 1 divided by e^-x.

• As e^x - 1.

• As e^-x.

• As e^x + 1.

What is the result of the limit of 3 divided by e^x - 3 as x approaches negative infinity?

• The result is 0.

• The result is 1.

• The result is -1.

• The result is 3.

What are the horizontal asymptotes of the function discussed in the video?

• y = 0 and y = 1

• y = 0 and y = -1

• y = 1 and y = 0

• y = 1 and y = -1

As x approaches positive infinity, what do the function values approach?

• They approach -1.

• They approach 0.

• They approach 1.

• They remain constant.

As x approaches negative infinity, what do the function values approach?

• They approach 1.

• They approach -1.

• They approach 0.

• They remain constant.

What does the graph of the function verify?

• The function has vertical asymptotes.

• The function has no asymptotes.

• The function has two horizontal asymptotes.

• The function has one horizontal asymptote.

What is the significance of the horizontal asymptotes in the graph?

• They represent the maximum values of the function.

• They indicate the points of intersection with the x-axis.

• They show the limits of the function as x approaches infinity.

• They indicate where the function is undefined.