Determine the following limits.
lim x→1^− x/ ln x

Verified step by step guidance

1. Identify the type of limit: This is a one-sided limit as \( x \to 1^- \), meaning we approach 1 from the left.
2. Substitute \( x = 1 \) into the expression \( \frac{x}{\ln x} \) to check if it results in an indeterminate form.
3. Recognize that as \( x \to 1^- \), \( \ln x \to \ln(1) = 0 \) and \( x \to 1 \), leading to the indeterminate form \( \frac{1}{0} \).
4. Apply L'Hôpital's Rule, which is used to evaluate limits of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Differentiate the numerator and the denominator separately.
5. Differentiate: The derivative of the numerator \( x \) is 1, and the derivative of the denominator \( \ln x \) is \( \frac{1}{x} \). Rewrite the limit as \( \lim_{x \to 1^-} \frac{1}{\frac{1}{x}} \).

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as x approaches 1 from the left, which requires analyzing the function's values as they get closer to this point.

Natural Logarithm (ln)

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a crucial function in calculus, particularly in limits and derivatives, as it has unique properties, such as being undefined for non-positive values. Understanding how ln(x) behaves as x approaches 1 is essential for evaluating the limit in the given question.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In the limit lim x→1^− x/ln x, both the numerator and denominator approach 0 as x approaches 1 from the left, creating an indeterminate form. Recognizing this allows us to apply techniques such as L'Hôpital's Rule to resolve the limit.