Understanding Limits of Natural Log Functions

This video tutorial covers how to find the limits of natural log functions, including examples with Ln X, Ln of absolute value of X, Ln of cosine X, and Ln of sine X. It explains the concept of vertical asymptotes, one-sided limits, and the importance of graphing to visualize limits. The tutorial also demonstrates the use of factorization and properties of logarithms to solve complex limit problems.

What is the behavior of ln(x) as x approaches 0 from the right?

• It oscillates.

• It remains constant.

• It approaches positive infinity.

• It approaches negative infinity.

Why does the limit of ln(x) as x approaches 0 from the left not exist?

• Because ln(x) is not defined for x ≤ 0.

• Because ln(x) has a horizontal asymptote at x=0.

• Because ln(x) is defined for all x.

• Because ln(x) is a linear function.

What is the limit of ln(|x|) as x approaches 0?

• Negative infinity

• Does not exist

• Zero

• Positive infinity

How does the graph of ln(|x|) differ from ln(x)?

• It is a reflection of ln(x) across the x-axis.

• It is a reflection of ln(x) across the y-axis.

• It is a horizontal shift of ln(x).

• It is a vertical shift of ln(x).

What is the limit of ln(cos(x)) as x approaches 0 from the left?

• 0

• Negative infinity

• Positive infinity

• 1

Why can direct substitution be used for ln(cos(x)) as x approaches 0 from the left?

• Because cos(x) approaches 1.

• Because cos(x) approaches 0.

• Because cos(x) approaches negative infinity.

• Because cos(x) is undefined at x=0.

What is the limit of ln(sin(x)) as x approaches 0 from the right?

• Negative infinity

• Zero

• Does not exist

• Positive infinity

Why is it important to know the direction from which y approaches zero in ln(sin(x))?

• To determine if the limit is a finite number.

• To determine if the limit is zero.

• To determine if the limit exists.

• To determine if the limit is positive or negative infinity.

What is the limit of ln(x^2 - 9) as x approaches 3 from the right?

• Positive infinity

• Zero

• Negative infinity

• Does not exist

What technique is used to evaluate the limit of ln(x^2 - 9) as x approaches 3 from the right?

• Graphical analysis

• Direct substitution

• Factoring and properties of logarithms

• Numerical approximation