Exploring Two Key Limits with the Squeeze Theorem

Why can't the two important limits be evaluated using algebraic methods?

• Because they result in indeterminate forms

• Because they involve complex numbers

• Because they are undefined

• Because they are irrational

What theorem is introduced to evaluate the two important limits?

• Mean Value Theorem

• Squeeze Theorem

• Intermediate Value Theorem

• Fundamental Theorem of Calculus

What is the limit of sin(x)/x as x approaches zero?

• 0

• 1

• Infinity

• Undefined

In the proof of the first limit, what geometric shape is used to compare areas?

• Square

• Triangle

• Rectangle

• Circle

What is the behavior of the function sin(x)/x as x approaches zero from both sides?

• It approaches 1

• It diverges

• It oscillates

• It approaches 0

What is the limit of (1 - cos(x))/x as x approaches zero?

• 1

• 0

• Infinity

• Undefined

Which algebraic manipulation is used in the proof of the second limit?

• Multiplying by the conjugate

• Rationalizing

• Simplifying complex fractions

• Factoring

What trigonometric identity is used in the proof of the second limit?

• cos(2x) = 2cos^2(x) - 1

• sin(2x) = 2sin(x)cos(x)

• tan(x) = sin(x)/cos(x)

• sin^2(x) + cos^2(x) = 1

In the application of limits, what substitution is made for sin(10x)/10x?

• x = 10

• Theta = x/10

• x = 1/10

• Theta = 10x

How is the limit of sin(5x)/x as x approaches zero evaluated?

• By direct substitution

• By factoring

• By introducing a factor of 5

• By using L'Hôpital's rule