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

lim_x→ 0 (eˣ - x - 1) / 5x²

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First, identify the form of the limit as x approaches 0. Substitute x = 0 into the expression (eˣ - x - 1) / 5x² to see 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 eˣ - x - 1 with respect to x. The derivative of eˣ is eˣ, the derivative of -x is -1, and the derivative of -1 is 0. So, the derivative of the numerator is eˣ - 1.

Differentiate the denominator 5x² with respect to x. The derivative of 5x² is 10x.

Apply l'Hôpital's Rule by taking the limit of the new expression (eˣ - 1) / 10x as x approaches 0. Substitute x = 0 again to check if the limit is still indeterminate. If it is, apply l'Hôpital's Rule again. Otherwise, evaluate the limit directly.

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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 where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

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 ∞/∞. The rule 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 separately. This process can be repeated if the result remains indeterminate.

Exponential Functions

Exponential functions, such as eˣ, are functions where a constant base is raised to a variable exponent. They are characterized by their rapid growth and unique properties, such as the fact that the derivative of eˣ is eˣ itself. Understanding the behavior of exponential functions is essential for evaluating limits involving them, especially as they approach specific values.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.lim_x→ 0 (eˣ - x - 1) / 5x²