Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x(e¹/ˣ − 1)

검증된 단계별 안내

Recognize that the limit involves the expression \(x \left(e^{1/x} - 1\right)\) as \(x\) approaches infinity, which suggests using the Taylor series expansion of the exponential function around 0.

Recall the Taylor series expansion for \(e^t\) around \(t=0\): \[e^t = 1 + t + \frac{t^2}{2!} + \frac{t^3}{3!} + \cdots\]

Substitute \(t = \frac{1}{x}\) into the series to get: \[e^{1/x} = 1 + \frac{1}{x} + \frac{1}{2x^2} + \frac{1}{6x^3} + \cdots\]

Rewrite the original expression using this expansion: \[x \left(e^{1/x} - 1\right) = x \left(\frac{1}{x} + \frac{1}{2x^2} + \frac{1}{6x^3} + \cdots \right)\]

Simplify the expression by multiplying \(x\) inside the parentheses and then analyze the behavior of each term as \(x \to \infty\) to determine the limit.

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주요 개념

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Limits at Infinity

Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how functions behave as x approaches infinity helps determine if the function approaches a finite value, infinity, or does not exist.

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 simplification of complex expressions, especially useful for evaluating limits.

Exponential Function and Its Expansion

The exponential function e^x can be expanded as a Taylor series: e^x = 1 + x + x²/2! + ... . For small values of x, this expansion helps approximate e^(1/x) and analyze the limit by substituting the series into the expression.

Limits Evaluate the following limits using Taylor series.lim ₓ→∞ x(e¹/ˣ − 1)