Question
8. Multiple Cholce 1 point Determine if L'Hopital's Rule can be used. If it can, solve the limit. If it cannot choose the appropriate response. \( \lim _{x \rightarrow 1} \frac{\ln x}{x-1} \) L'Hopital's Rule does not apply. (1) Previous
Ask by Mejia Colon. in the United States
Feb 26,2025
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Answer
L'Hôpital's Rule can be used. The limit is 1.
Solution
We first check if the limit is in an indeterminate form. As x approaches 1, the numerator ln(x) tends to ln(1) = 0 and the denominator x - 1 tends to 0. Since both numerator and denominator approach 0, we have a 0/0 indeterminate form, meaning L'Hôpital's Rule can be used.
Applying L'Hôpital's Rule, we differentiate the numerator and denominator:
• Derivative of ln(x) is 1/x.
• Derivative of (x - 1) is 1.
So, the limit becomes:
limₓ→₁ (1/x) = 1/1 = 1
Thus, the limit is 1.
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Simplify this solution Extra Insights
L'Hôpital's Rule is perfect for situations where we encounter indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). In this case, as \( x \) approaches 1, both \( \ln x \) and \( x - 1 \) approach 0, making it a \( \frac{0}{0} \) form. So, we can apply L'Hôpital's Rule! When we differentiate the numerator and denominator, we get \( \frac{\frac{1}{x}}{1} \). Now substituting \( x = 1 \) gives us \( \frac{1}{1} = 1 \). Voilà! The limit is 1.
