Evaluate $\lim _{x \rightarrow 0} \frac{\ln (1-x)}{e^x-1}=$
Answer (2)
The limit lim x → 0 e x − 1 l n ( 1 − x ) evaluates to − 1 using L'Hôpital's Rule after recognizing it as an indeterminate form. By differentiating both the numerator and the denominator, and subsequently taking the limit, we find the solution. Hence, the final result is − 1 .
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Recognize that the limit is in the indeterminate form 0 0 .
Apply L'Hôpital's Rule by differentiating the numerator and the denominator.
Find the derivative of the numerator: d x d ln ( 1 − x ) = 1 − x − 1 .
Find the derivative of the denominator: d x d ( e x − 1 ) = e x .
Evaluate the limit of the derivatives: lim x → 0 e x 1 − x − 1 = − 1 .
The final answer is − 1 .
Explanation
Problem Analysis We are asked to evaluate the limit of the function e x − 1 l n ( 1 − x ) as x approaches 0 .
Indeterminate Form The limit is of the indeterminate form 0 0 as x approaches 0 , since ln ( 1 − 0 ) = ln ( 1 ) = 0 and e 0 − 1 = 1 − 1 = 0 .
L'Hôpital's Rule Since the limit is of the indeterminate form 0 0 , we can apply L'Hôpital's Rule, which states that if lim x → c g ( x ) f ( x ) is of the form 0 0 or ∞ ∞ , then lim x → c g ( x ) f ( x ) = lim x → c g ′ ( x ) f ′ ( x ) , provided the limit exists.
Differentiate Numerator Differentiate the numerator with respect to x : d x d ln ( 1 − x ) = 1 − x 1 ⋅ d x d ( 1 − x ) = 1 − x 1 ⋅ ( − 1 ) = 1 − x − 1 .
Differentiate Denominator Differentiate the denominator with respect to x : d x d ( e x − 1 ) = e x .
Apply L'Hôpital's Rule Apply L'Hôpital's Rule: x → 0 lim e x − 1 ln ( 1 − x ) = x → 0 lim e x 1 − x − 1 = x → 0 lim ( 1 − x ) e x − 1 .
Evaluate the Limit Evaluate the limit: x → 0 lim ( 1 − x ) e x − 1 = ( 1 − 0 ) e 0 − 1 = ( 1 ) ( 1 ) − 1 = − 1.
Final Answer Therefore, the limit is x → 0 lim e x − 1 ln ( 1 − x ) = − 1.
Examples
L'Hôpital's Rule is not just a theoretical concept; it's incredibly useful in various fields. For instance, in physics, when analyzing the behavior of circuits or systems as they approach certain critical points, you often encounter indeterminate forms. Applying L'Hôpital's Rule helps simplify these complex scenarios, allowing engineers and physicists to make accurate predictions and design more efficient systems. It's also used in economics to analyze growth models and determine equilibrium points in markets.
