Limits and Trigonometric Functions

The video tutorial explores trigonometric relationships, focusing on the identities involving sine and cosine. It discusses the impact of dividing by sine on inequalities and introduces the concept of limits, particularly for small values of x. The Squeeze Theorem is explained in detail, demonstrating how it applies to trigonometric functions. The tutorial concludes with examples of applying limits to different trigonometric expressions, emphasizing the importance of understanding these concepts in calculus.

What is the basic relationship between sine and cosine in trigonometry?

• Sine and cosine are equal

• Sine divided by cosine equals tangent

• Sine is the reciprocal of cosine

• Sine is the tangent of cosine

When dividing through by sine x, why does the direction of the inequality remain unchanged?

• Because sine x is a constant

• Because sine x is always negative

• Because sine x is zero

• Because sine x is positive for small x

What mathematical concept is used to describe values approaching zero?

• Derivatives

• Integrals

• Limits

• Functions

What happens to the value of 1 as x approaches zero?

• It becomes negative

• It becomes infinite

• It remains unchanged

• It becomes zero

What is the Squeeze Theorem used to demonstrate in this context?

• That x over sine x approaches one

• That x over sine x approaches infinity

• That x over sine x approaches negative one

• That x over sine x approaches zero

How does the Squeeze Theorem apply to the limit of x over tan x?

• It shows that the limit is negative one

• It shows that the limit is infinite

• It shows that the limit is one

• It shows that the limit is zero

What can be done to simplify the limit of 2x over sine x?

• Multiply by sine x

• Add 2 to the expression

• Subtract 2 from the expression

• Factor out the 2

What adjustment is made to x over sine half x to apply the Squeeze Theorem?

• Multiply by 2

• Divide by 2

• Add 1

• Subtract 1

Why can't the half be factored out directly in x over sine half x?

• Because x is not a constant

• Because sine is not a linear function

• Because the expression is already simplified

• Because half is not a factor

What is the result of the limit of x over sine x as x approaches zero?

• Zero

• Infinity

• Negative one

• One