17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)

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First, identify the form of the limit as x approaches 0. Substitute x = 0 into the expression to check if it results in an indeterminate form like 0/0.

Since substituting x = 0 gives 0/0, l'Hôpital's Rule is applicable. This rule states that if the limit of f(x)/g(x) as x approaches a value results in 0/0 or ∞/∞, then the limit can be found by differentiating the numerator and the denominator separately.

Differentiate the numerator: The derivative of eˣ is eˣ, and the derivative of sin x is cos x. Therefore, the derivative of the numerator eˣ - sin x - 1 is eˣ - cos x.

Differentiate the denominator: The derivative of x⁴ is 4x³, the derivative of 8x³ is 24x², and the derivative of 12x² is 24x. Therefore, the derivative of the denominator x⁴ + 8x³ + 12x² is 4x³ + 24x² + 24x.

Apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→0 (eˣ - cos x) / (4x³ + 24x² + 24x). Evaluate this new limit by substituting x = 0 again, and if necessary, apply l'Hôpital's Rule repeatedly until the limit can be determined.

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주요 개념

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this problem, evaluating the limit as x approaches 0 helps determine the behavior of the function near that point.

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits in complex expressions.

Taylor Series Expansion

The Taylor Series Expansion is a way to represent functions as infinite sums of terms calculated from the values of their derivatives at a single point. For functions like eˣ and sin x, their Taylor series can be used to approximate their values near x = 0, which is useful for simplifying the limit expression in this problem.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→ 0 (eˣ - sin x - 1) / (x⁴ + 8x³ + 12x²)