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


lim_x→∞ (e¹/ₓ - 1)/(1/x)

Verified step by step guidance

1. First, identify the form of the limit as x approaches infinity. Substitute x with infinity in the expression (e^(1/x) - 1)/(1/x) to see if it results in an indeterminate form like 0/0 or ∞/∞.
2. Notice that as x approaches infinity, 1/x approaches 0. Therefore, e^(1/x) approaches e^0, which is 1. This makes the numerator e^(1/x) - 1 approach 0, and the denominator 1/x also approaches 0, resulting in the indeterminate form 0/0.
3. Since the limit is in the indeterminate form 0/0, l'Hôpital's Rule can be applied. According to l'Hôpital's Rule, take the derivative of the numerator and the derivative of the denominator separately.
4. The derivative of the numerator e^(1/x) - 1 with respect to x is found using the chain rule. The derivative of e^(1/x) is e^(1/x) multiplied by the derivative of 1/x, which is -1/x². Therefore, the derivative of the numerator is -e^(1/x)/x².
5. The derivative of the denominator 1/x with respect to x is -1/x². Now, apply l'Hôpital's Rule by substituting these derivatives into the limit: lim_x→∞ (-e^(1/x)/x²)/(-1/x²). Simplify the expression and evaluate the limit as x approaches infinity.

Key Concepts

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 at points where it may not be explicitly defined, such as at infinity or discontinuities. 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 e^(1/x), are functions where a constant base is raised to a variable exponent. They exhibit unique properties, particularly as the variable approaches infinity or zero. Understanding the behavior of exponential functions is essential for evaluating limits involving them, especially in the context of growth rates and asymptotic behavior.