Understanding Limits and Continuity

The video tutorial explains how to determine the value of f(6) given that f(x) and g(x) are continuous functions, with g(6) = -8, and the limit of the sum of 5 times f(x) and f(x) times g(x) as x approaches 6 equals -14. The solution involves applying limit laws, performing direct substitution, and solving the resulting equation to find f(6) = 14/3.

What is the value of g(6) given in the problem?

• 6

• -8

• 0

• 8

What is the initial limit expression given in the problem?

• Limit of f(x)g(x) as x approaches 6

• Limit of g(x) as x approaches 6

• Limit of the sum of 5f(x) and f(x)g(x) as x approaches 6

• Limit of f(x) as x approaches 6

According to limit laws, how can the limit of a sum be expressed?

• As a product of limits

• As a sum of limits

• As a difference of limits

• As a quotient of limits

What does the limit of a product equal according to limit properties?

• A product of limits

• A sum of limits

• A quotient of limits

• A difference of limits

Why can direct substitution be used in this problem?

• Because f(x) and g(x) are integrable

• Because f(x) and g(x) are discrete

• Because f(x) and g(x) are differentiable

• Because f(x) and g(x) are continuous

What equation is formed after substituting g(6) into the limit expression?

• 5f(6) - f(6) = -14

• 5f(6) + f(6) = -14

• 5f(6) - 8f(6) = -14

• 5f(6) + 8f(6) = -14

What is the result of combining like terms in the equation 5f(6) - 8f(6)?

• -13f(6)

• 13f(6)

• 3f(6)

• -3f(6)

What is the final value of f(6) after solving the equation?

• 3/14

• 14/3

• -3/14

• -14/3

What operation is performed to isolate f(6) in the equation -3f(6) = -14?

• Multiplying both sides by 3

• Dividing both sides by -3

• Subtracting 3 from both sides

• Adding 3 to both sides

What is the significance of the continuity of f(x) and g(x) in this problem?

• It allows for differentiation

• It allows for direct substitution

• It allows for approximation

• It allows for integration