Understanding Limits Involving Trigonometric Functions

This video tutorial explores determining limits involving trigonometric functions using various techniques. It covers numerical, graphical, and analytical approaches, and provides an informal definition of limits. The video also discusses conditions under which limits do not exist, such as differing values from the left and right, unbounded increase or decrease, and oscillation. Examples include limits of sin(x) as x approaches π/6, tan(x) as x approaches π/2, and sin(1/x) as x approaches 0, illustrating both existent and non-existent limits.

What are the three main approaches to finding limits?

• Numerical, Graphical, Analytical

• Visual, Logical, Empirical

• Algebraic, Geometric, Statistical

• Theoretical, Practical, Experimental

According to the informal definition, when does the limit of f(x) as x approaches c equal L?

• When f(x) is greater than L

• When f(x) values are close to L for x near c

• When f(x) is undefined at c

• When f(x) is always equal to L

Which of the following is NOT a reason for the non-existence of a limit?

• Function is continuous at c

• Function oscillates between fixed values

• Function increases or decreases without bound

• Function approaches different values from left and right

What is the limit of sin(x) as x approaches π/6?

• π/6

• 1

• 1/2

• 0

Why can direct substitution be used to find the limit of sin(x) as x approaches π/6?

• Because sin(x) is discontinuous at π/6

• Because sin(x) is undefined at π/6

• Because sin(x) is smooth and continuous around π/6

• Because sin(x) is a constant function

What happens to the limit of tan(x) as x approaches π/2?

• It approaches zero

• It approaches infinity

• It does not exist

• It approaches a constant value

Why does the limit of tan(x) not exist as x approaches π/2?

• Because tan(x) is continuous at π/2

• Because tan(x) approaches different values from left and right

• Because tan(x) is a constant function

• Because tan(x) is zero at π/2

What is the behavior of sin(1/x) as x approaches zero?

• It approaches a fixed value

• It oscillates between -1 and +1

• It becomes undefined

• It approaches zero

Why does the limit of sin(1/x) not exist as x approaches zero?

• Because it is a constant function

• Because it is undefined at zero

• Because it approaches a single value

• Because it oscillates faster and faster between -1 and +1

What does the graph of sin(1/x) show as x approaches zero?

• A straight line

• A parabolic curve

• A constant value

• An oscillating pattern