Understanding Limits at Infinity

This video tutorial covers the concept of limits at infinity, explaining how to determine limits as inputs grow larger or smaller without bound. It discusses infinite limits, horizontal asymptotes, and provides guidelines for finding limits of rational functions. The tutorial includes examples to illustrate these concepts, using graphs and tables to demonstrate how function values behave as they approach infinity or negative infinity.

What does it mean when we say a function has a limit at infinity?

• The function's values decrease to negative infinity.

• The function's values become constant.

• The function's values oscillate indefinitely.

• The function's values approach a specific value as inputs grow larger or smaller without bound.

According to the theorem discussed, what happens to the limit of C/x^r as x approaches infinity?

• It becomes undefined.

• It approaches infinity.

• It approaches zero.

• It approaches a constant value.

How is a horizontal asymptote related to limits at infinity?

• It indicates where the function's values become undefined.

• It shows the value that the function's values approach as x approaches infinity or negative infinity.

• It represents the maximum value of the function.

• It is unrelated to limits at infinity.

What is the general guideline for finding limits at infinity of rational functions?

• Add a constant to each term.

• Multiply each term by the highest power of x in the numerator.

• Subtract the highest power of x from each term.

• Divide each term by the highest power of x in the denominator.

When using the shortcut method, what is the limit if the degree of the denominator is greater than the degree of the numerator?

• The limit is zero.

• The limit is infinity.

• The limit is the ratio of the leading coefficients.

• The limit does not exist.

In the example of 1/x + 3, what is the limit as x approaches infinity?

• Infinity

• 0

• 1

• 3

For the function 2x + 5 / (x - 1), what is the limit as x approaches negative infinity?

• 2

• Infinity

• 0

• 1

What is the limit of 3x^2 - 5x / (x^3 + 7) as x approaches infinity?

• 3

• 1

• 0

• Infinity

In the example of 2x^2 - 5x / 4x, what is the limit as x approaches infinity?

• 0

• 1

• Infinity

• 2

Why does the limit of 2x^2 - 5x / 4x not exist as x approaches infinity?

• Because both the numerator and denominator grow at the same rate.

• Because the numerator grows faster than the denominator.

• Because the denominator grows faster than the numerator.

• Because the function oscillates indefinitely.