Understanding Limits of Composite Functions

This video tutorial explores the concept of determining the limit of a composite function when the limit of the inner function does not exist. It explains that the non-existence of the inner function's limit does not necessarily imply the non-existence of the composite function's limit. The tutorial uses graphs to analyze the left and right-side limits of the composite function f(G(x)) as x approaches 2. It concludes that if both one-sided limits are equal, the limit of the composite function exists, even if the inner function's limit does not.

What is the main focus when determining the limit of a composite function?

• The limit of the composite function

• The limit of both inner and outer functions

• The limit of the inner function

• The limit of the outer function

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CCSS.HSF-IF.C.7E

What does the discontinuity at x = 2 in G(x) indicate?

• The function is continuous

• The limit exists

• The limit does not exist

• The function is differentiable

Tags

CCSS.HSF-IF.C.7E

When approaching x = 2 from the right in G(x), what y-value is approached?

• Negative 4

• Positive 3

• Positive 4

• Negative 3

Tags

CCSS.HSF-IF.C.7E

How is the right-side limit of f(G(x)) determined?

• By evaluating f(x) at x = 2

• By evaluating G(x) at x = 4

• By evaluating f(x) at x = 4

• By evaluating G(x) at x = 2

Tags

CCSS.HSF-IF.C.7E

What is the y-value approached by f(x) as x approaches 4 from the left?

• Positive 5

• Positive 4

• Positive 3

• Positive 6

Tags

CCSS.HSF-IF.C.7E

What y-value does G(x) approach from the left as x approaches 2?

• Positive 5

• Positive 4

• Positive 3

• Positive 2

Tags

CCSS.HSF-IF.C.7E

How is the left-side limit of f(G(x)) determined?

• By evaluating f(x) at x = 3

• By evaluating G(x) at x = 2

• By evaluating G(x) at x = 3

• By evaluating f(x) at x = 2

Tags

CCSS.HSF-IF.C.7E

What is the y-value approached by f(x) as x approaches 3 from the left?

• Positive 6

• Positive 3

• Positive 4

• Positive 5

What conclusion is drawn about the limit of the composite function?

• It equals 4

• It equals 3

• It does not exist

• It equals 5

Why does the limit of the composite function exist even if the inner function's limit does not?

• Because the composite function is undefined

• Because the inner function is differentiable

• Because the outer function is continuous

• Because the one-sided limits are equal