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

lim_x→0⁺ x²ˣ

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Identify the form of the limit as x approaches 0 from the positive side. The expression x²ˣ can be rewritten as e^(x ln(x²)). As x approaches 0⁺, x ln(x²) approaches an indeterminate form 0 * (-∞).

Rewrite the expression x²ˣ as e^(2x ln(x)). This helps in applying the limit to the exponent first, which is 2x ln(x).

Consider the limit of the exponent: lim_(x→0⁺) 2x ln(x). This is an indeterminate form of type 0 * (-∞), which can be rewritten as lim_(x→0⁺) (ln(x) / (1/(2x))).

Apply l'Hôpital's Rule to the limit of the exponent, since it is in the indeterminate form ∞/∞. Differentiate the numerator and the denominator separately: the derivative of ln(x) is 1/x, and the derivative of 1/(2x) is -1/(2x²).

Evaluate the new limit after applying l'Hôpital's Rule: lim_(x→0⁺) (1/x) / (-1/(2x²)) simplifies to lim_(x→0⁺) -2x. As x approaches 0⁺, this limit approaches 0. Therefore, the original limit of x²ˣ is e^0, which simplifies to 1.

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주요 개념

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Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or infinity. Evaluating limits is essential for defining derivatives and integrals, which are core components of calculus.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. In the context of the limit lim_x→0⁺ x²ˣ, the expression involves an exponential function with a variable base. Understanding the properties of exponential functions, especially their behavior as the exponent approaches zero, is crucial for evaluating this limit.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits, especially when direct substitution is not possible.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→0⁺ x²ˣ