Limits Evaluate the following limits using Taylor series.
lim ₓ→₁ (x 1)/(ln x)

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Identify the limit expression: \(\lim_{x \to 1} \frac{x - 1}{\ln x}\).

Recognize that both numerator and denominator approach 0 as \(x \to 1\), indicating an indeterminate form \(\frac{0}{0}\), which suggests using a series expansion or L'Hôpital's Rule.

Expand \(\ln x\) around \(x = 1\) using the Taylor series: \(\ln x = (x - 1) - \frac{(x - 1)^2}{2} + \frac{(x - 1)^3}{3} - \cdots\).

Substitute the Taylor series expansion of \(\ln x\) into the limit expression to get \(\frac{x - 1}{(x - 1) - \frac{(x - 1)^2}{2} + \cdots}\).

Simplify the fraction by factoring out \((x - 1)\) in the denominator and canceling it with the numerator, then evaluate the limit as \(x \to 1\) by considering the leading terms.

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Limits and Indeterminate Forms

Limits describe the behavior of a function as the input approaches a particular value. When direct substitution leads to expressions like 0/0 or ∞/∞, these are called indeterminate forms, requiring special techniques such as series expansions to evaluate the limit.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. It approximates functions near that point, allowing complex expressions to be simplified and limits to be evaluated by examining leading terms.

Logarithmic Function Behavior Near 1

The natural logarithm function ln(x) behaves predictably near x = 1, where ln(1) = 0. Its Taylor series expansion around x = 1 starts as ln(x) ≈ (x - 1) - (x - 1)^2/2 + ..., which helps in simplifying limits involving ln(x) near 1.

Limits Evaluate the following limits using Taylor series.lim ₓ→₁ (x 1)/(ln x)