Use series to evaluate the limits in Exercises 29–40.
29. lim (x → 0) (e^x - (1 + x)) / x²

검증된 단계별 안내

Recall the Maclaurin series expansion for the exponential function \(e^x\), which is given by: \[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\]

Substitute the series expansion of \(e^x\) into the given limit expression: \[\lim_{x \to 0} \frac{e^x - (1 + x)}{x^2} = \lim_{x \to 0} \frac{\left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\right) - (1 + x)}{x^2}\]

Simplify the numerator by canceling out the \(1\) and \(x\) terms: \[\lim_{x \to 0} \frac{\frac{x^2}{2!} + \frac{x^3}{3!} + \cdots}{x^2}\]

Divide each term in the numerator by \(x^2\): \[\lim_{x \to 0} \left( \frac{1}{2!} + \frac{x}{3!} + \frac{x^2}{4!} + \cdots \right)\]

Evaluate the limit as \(x\) approaches 0 by substituting \(x = 0\) into the simplified expression, which leaves only the constant term(s).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. For functions like e^x, the series expansion around x=0 (Maclaurin series) helps approximate the function using polynomials, which simplifies limit evaluation.

Limit Evaluation Using Series

By substituting the series expansion into the limit expression, complex functions can be approximated by polynomials. This allows direct simplification and evaluation of limits, especially when the original expression results in indeterminate forms like 0/0.

Handling Indeterminate Forms

Limits that yield forms like 0/0 require algebraic manipulation or series expansion to resolve. Using series expansions helps rewrite the numerator and denominator to identify the leading terms, enabling the determination of the limit's value.

Use series to evaluate the limits in Exercises 29–40.29. lim (x → 0) (e^x - (1 + x)) / x²