Functions and Limits Analysis

The video tutorial explains the order of functions f(x), g(x), and h(x) near x=2, using graphs to illustrate that f(x) is less than or equal to g(x), which is less than or equal to h(x). It introduces the definitions of these functions and analyzes their limits as x approaches 2, applying the squeeze theorem to conclude that all functions approach the value 1. The tutorial concludes with a verification of the results.

What is the purpose of creating a compound inequality for the functions f(x), g(x), and h(x)?

• To compare the functions at x = 0

• To find the maximum value of the functions

• To order the values of the functions for x-values near 2

• To determine the exact values of the functions at x = 2

For x = 3, which function has the smallest value?

• All have the same value

• h(x)

• g(x)

• f(x)

What is the relationship between f(x), g(x), and h(x) for the x-values depicted in the graph?

• f(x) > g(x) > h(x)

• f(x) < g(x) < h(x)

• f(x) = g(x) = h(x)

• f(x) > h(x) > g(x)

What is the definition of f(x) in terms of its components?

• 2x + 1

• 2 times the square root of (x - 1) minus 1

• e to the x minus 2

• x squared minus 1

Which function is represented by the expression e to the x minus 2?

• f(x)

• h(x)

• None of the above

• g(x)

What theorem is used to determine the limit of g(x) as x approaches 2?

• Mean Value Theorem

• Fundamental Theorem of Calculus

• Intermediate Value Theorem

• Squeeze Theorem

What is the limit of f(x) as x approaches 2?

• Undefined

• 2

• 0

• 1

What is the principal root of 2 minus 1?

• 0

• 1

• 2

• 3

What is the final conclusion about the limit of g(x) as x approaches 2?

• It is undefined

• It is greater than 1

• It is equal to 1

• It is less than 1

Why must g(x) approach 1 as x approaches 2?

• Because f(x) and h(x) both approach 1

• Because g(x) is always 1

• Because x is equal to 2

• Because g(x) is undefined at x = 2