Using L'Hôpital's Rule, we differentiate the numerator and denominator to find lim (x -> 0) (1/(1 + x)) = 1.
Questions & Step-by-step Solutions
1 item
Q
Q: What is the limit: lim (x -> 0) (ln(1 + x)/x)?
Solution: Using L'Hôpital's Rule, we differentiate the numerator and denominator to find lim (x -> 0) (1/(1 + x)) = 1.
Steps: 8
Step 1: Identify the limit we want to find: lim (x -> 0) (ln(1 + x)/x).
Step 2: Check if we can directly substitute x = 0 into the expression. If we do, we get ln(1 + 0)/0, which is ln(1)/0 = 0/0. This is an indeterminate form.
Step 3: Since we have an indeterminate form (0/0), we can use L'Hôpital's Rule. This rule states that we 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: Now we rewrite the limit using the derivatives: lim (x -> 0) (1/(1 + x)/1).
Step 7: Substitute x = 0 into the new expression: 1/(1 + 0) = 1/1 = 1.
