Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)

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Question: Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)

Options:

Correct Answer: 1

Solution:

Using L\'Hôpital\'s Rule, differentiate the numerator and denominator: lim (x -> 0) (1/(1 + x))/(1) = 1.

Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)

Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)

Step 1: Identify the limit we want to evaluate: lim (x -> 0) (ln(1 + x)/x).
Step 2: Check if the limit is in an indeterminate form. As x approaches 0, ln(1 + x) approaches ln(1) which is 0, and x approaches 0. So we have 0/0, which is indeterminate.
Step 3: Apply L'Hôpital's Rule, which states that if you have an indeterminate form like 0/0, you can take the derivative of the numerator and the derivative of the denominator.
Step 4: Differentiate the numerator: The derivative of ln(1 + x) is 1/(1 + x).
Step 5: Differentiate the denominator: The derivative of x is 1.
Step 6: Rewrite the limit using the derivatives: lim (x -> 0) (1/(1 + x))/1.
Step 7: Simplify the expression: lim (x -> 0) (1/(1 + x)).
Step 8: Substitute x = 0 into the simplified expression: 1/(1 + 0) = 1.
Step 9: Conclude that the limit is 1.

Limits and Continuity – Understanding how to evaluate limits, especially those that result in indeterminate forms.
L'Hôpital's Rule – Applying L'Hôpital's Rule to resolve limits that yield 0/0 or ∞/∞ forms.
Natural Logarithm Properties – Utilizing properties of logarithms to simplify expressions before taking limits.