Limits and Rational Functions

This video tutorial provides additional examples on determining limits at infinity, focusing on more challenging cases than the previous video. It reviews major ideas, such as when the numerator is constant and the denominator is a power of x, leading to a limit of zero. The video outlines general guidelines and shortcuts for finding limits of rational functions, including cases where the degrees of the numerator and denominator differ. Three examples are explored: a rational function, a sine function, and an arctangent function, each demonstrating different approaches to finding limits at infinity.

What is the limit of a function where the numerator is a constant and the denominator is x raised to a positive rational power as x approaches infinity?

• The limit is 0

• The limit is 1

• The limit does not exist

• The limit is infinity

If the degree of the numerator is less than the degree of the denominator in a rational function, what is the limit as x approaches infinity?

• The limit is 0

• The limit is 1

• The limit is infinity

• The limit does not exist

When the degrees of the numerator and denominator are equal, what determines the limit of the rational function as x approaches infinity?

• The difference of the coefficients

• The sum of the coefficients

• The product of the coefficients

• The ratio of the leading coefficients

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

• The limit does not exist

• The limit is infinity

• The limit is 1

• The limit is 0

In Example 1, what is the limit of (2x + 1) / sqrt(x^2 - x) as x approaches infinity?

• The limit is 2

• The limit is 1

• The limit does not exist

• The limit is 0

In Example 1, what method is used to simplify the expression (2x + 1) / sqrt(x^2 - x)?

• Multiplying by the conjugate

• Dividing by the highest power of x

• Using L'Hôpital's rule

• Completing the square

In Example 2, what is the limit of sin(1/x) as x approaches infinity?

• The limit is 1

• The limit is infinity

• The limit is 0

• The limit does not exist

In Example 2, what graphical feature helps verify the limit of sin(1/x) as x approaches infinity?

• Vertical asymptote

• Horizontal asymptote

• Point of inflection

• Local maximum

In Example 3, what is the limit of 5/x - arctan(x) as x approaches infinity?

• The limit is 0

• The limit is -π/2

• The limit is π/2

• The limit does not exist

In Example 3, what value does arctan(x) approach as x approaches infinity?

• π

• π/2

• π/4

• 0