Understanding Limits of Inverse Trigonometric Functions

The video tutorial explores the concept of limits involving inverse trigonometric functions. It begins with the limit of the inverse tangent function as x approaches infinity and negative infinity, highlighting the importance of understanding the graph's asymptotes. The tutorial then examines the limit of arc tangent as x approaches 2, emphasizing the need for one-sided limits due to a vertical asymptote. Next, it analyzes the limit of arc cosine of 1/x as x approaches infinity, using the graph's domain and range. Finally, the video evaluates the limit of an arc sine expression involving a rational function, demonstrating the use of trigonometric identities and graph analysis.

What is the limit of the inverse tangent function as x approaches positive infinity?

• pi/2

• 0

• Negative pi/2

• Infinity

As x approaches negative infinity, what does the inverse tangent function approach?

• 0

• pi/2

• Negative pi/2

• Infinity

Why can't direct substitution be used to find the limit of the inverse tangent function as x approaches 2?

• The function is continuous at x=2

• The function is undefined at x=2

• The denominator becomes zero

• The numerator becomes zero

What is the behavior of the function 1/(x-2) as x approaches 2 from the left?

• Remains constant

• Approaches negative infinity

• Approaches positive infinity

• Approaches zero

What is the limit of the inverse cosine function as x approaches infinity?

• 0

• pi/2

• pi

• Infinity

What is the domain of the inverse cosine function?

• 0 to 2pi

• Negative 1 to 1

• 0 to pi

• Negative infinity to infinity

How do you simplify the expression 5 + 2x^3 / (4x^3 - 8) as x approaches infinity?

• Subtract x^3

• Divide by x^3

• Add x^3

• Multiply by x^3

What is the arcsine of 1/2?

• pi/4

• pi/2

• pi/6

• pi/3

What is the range of the inverse sine function?

• Negative pi/2 to pi/2

• 0 to 2pi

• Negative pi to pi

• 0 to pi

In which quadrants does the inverse sine function exist?

• Quadrants I and IV

• Quadrants I and II

• Quadrants II and III

• Quadrants III and IV