Understanding the Squeeze Theorem and Its Application

The video tutorial explains the squeeze theorem and uses it to prove that the limit of sine x over x as x approaches 0 is 1. The instructor provides a visual proof using the unit circle, comparing areas of triangles and sectors to establish inequalities. The proof is supported by graphical representation, demonstrating the concept visually and confirming the limit using the squeeze theorem.

What is the main goal of using the squeeze theorem in this lesson?

• To calculate the area of a circle

• To find the derivative of sine of x

• To demonstrate the limit of sine of x over x as x approaches 0

• To prove the limit of cosine of x over x as x approaches 0

In the unit circle, what does the sine of an angle represent?

• The circumference of the circle

• The x-coordinate of a point on the circle

• The y-coordinate of a point on the circle

• The radius of the circle

How is the tangent of an angle defined in the context of the unit circle?

• As the ratio of the opposite side to the adjacent side

• As the ratio of the hypotenuse to the opposite side

• As the ratio of the radius to the circumference

• As the ratio of the adjacent side to the hypotenuse

What is the area of a sector in the unit circle with angle x radians?

• x/2

• x/4

• x/3

• x/π

Why do we take the absolute value of sine and tangent in the inequalities?

• To avoid division by zero

• To make the graph look better

• To simplify the calculation

• To ensure the values are always positive

What happens to the inequality signs when you invert both sides of an inequality?

• They disappear

• They switch direction

• They remain the same

• They become equal signs

What is the limit of the function 1 as x approaches 0?

• 1

• Undefined

• Infinity

• 0

What is the limit of cosine of x as x approaches 0?

• 1

• 0

• Undefined

• Infinity

How does the squeeze theorem help in proving the limit of sine of x over x?

• By showing it is always greater than 1

• By demonstrating it is always less than 0

• By proving it is equal to cosine of x

• By bounding it between two functions that both approach 1

What does the graphical representation show about sine of x over x?

• It is always above 1

• It diverges as x approaches 0

• It is always below cosine of x

• It is squeezed between 1 and cosine of x