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

Verified step by step guidance

1. 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.
2. Recall the Taylor series expansion for \(e^t\) around \(t=0\): \[e^t = 1 + t + \frac{t^2}{2!} + \frac{t^3}{3!} + \cdots\]
3. Substitute \(t = \frac{1}{x}\) into the series to get: \[e^{1/x} = 1 + \frac{1}{x} + \frac{1}{2x^2} + \frac{1}{6x^3} + \cdots\]
4. Rewrite the original expression using this expansion: \[x \left(e^{1/x} - 1\right) = x \left(\frac{1}{x} + \frac{1}{2x^2} + \frac{1}{6x^3} + \cdots \right)\]
5. 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.

Key Concepts

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.