Understanding Limits Involving Trigonometric Functions

The video tutorial explores the limit of cosine x over x squared minus one as x approaches infinity. It begins by reasoning that the numerator, cosine x, oscillates between -1 and 1, while the denominator grows infinitely large, leading the limit to approach zero. A more mathematical approach is then presented using inequalities to show that the limit is bounded between zero and zero, confirming the result. The tutorial concludes by reinforcing the understanding that the limit is zero.

What is the main problem discussed in the video?

• Finding the limit of cosine of x over x squared minus one.

• Finding the limit of cosine of x over x cubed minus one.

• Finding the limit of tangent of x over x squared minus one.

• Finding the limit of sine of x over x squared minus one.

How does the numerator, cosine of x, behave as x changes?

• It oscillates between -1 and 1.

• It increases linearly.

• It decreases linearly.

• It remains constant.

What happens to the denominator, x squared minus one, as x becomes very large?

• It approaches zero.

• It remains constant.

• It becomes infinitely large.

• It oscillates between -1 and 1.

What is the result of dividing a bounded numerator by an infinitely large denominator?

• The result approaches infinity.

• The result approaches zero.

• The result oscillates between -1 and 1.

• The result remains constant.

What is the maximum value that cosine of x can take?

• -1

• 1

• 0

• 2

What is the minimum value that cosine of x can take?

• 1

• 2

• -1

• 0

What inequality is used to show the limit approaches zero?

• 0 < limit < 1

• -1 < limit < 1

• -1 <= limit <= 1

• 0 <= limit <= 0

What is the final conclusion about the limit of the function as x approaches infinity?

• The limit is 1.

• The limit is infinity.

• The limit is zero.

• The limit is -1.

Why does the limit of cosine x over x squared minus one approach zero?

• Because both the numerator and denominator are unbounded.

• Because the denominator is bounded.

• Because the numerator is unbounded.

• Because the numerator is bounded and the denominator is unbounded.

What mathematical concept is primarily used in this video to find the limit?

• Differentiation

• Integration

• Inequalities

• Series expansion