Understanding Limits and Simplification Techniques

The video tutorial explains how to determine the limit of a function using algebraic techniques. It begins by discussing the limitations of direct substitution when faced with an indeterminate form. The tutorial then demonstrates how to factor and simplify the function using trigonometric identities, allowing for the determination of the limit through substitution. The process is verified graphically and with a table of values, confirming the limit as 1.5 or three halves.

What is the main challenge when trying to find the limit of the given function using direct substitution?

• The function is undefined for all x.

• The function is not continuous.

• The function has no limit.

• The function results in an indeterminate form.

Which algebraic technique is suggested to simplify the given function?

• Differentiation

• Using the quadratic formula

• Factoring

• Completing the square

What is the greatest common factor in the numerator of the given function?

• 2 cosine squared x

• 3

• 3 sine squared x

• 2

Which trigonometric identity is used to simplify the function?

• sine squared theta plus cosine squared theta equals one

• tangent squared theta plus one equals secant squared theta

• one minus sine squared theta equals cosine squared theta

• cosine squared theta minus sine squared theta equals one

After simplification, what is the expression for the function?

• 3 cosine squared x divided by 2

• 2 sine squared x divided by 3

• 3 sine squared x divided by 2

• 2 cosine squared x divided by 3

What is the value of cosine pi used in the final calculation of the limit?

• 0

• 1

• -1

• 0.5

What is the final value of the limit after simplification and substitution?

• 1

• 2

• 3

• 1.5

What is the approximate value of pi used in the table of values?

• 3.14

• 3.50

• 3.00

• 3.75

What does the graph of the original function show at pi radians?

• A continuous point

• A removable discontinuity

• A horizontal asymptote

• A vertical asymptote

What is the function value as x approaches pi from both sides?

• 1.5

• 1

• 0.5

• 2