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


lim_z→∞ (1 + 10/z²)ᶻ²

Verified step by step guidance

1. First, recognize that the limit lim_{z→∞} (1 + 10/z²)ᶻ² is of the form (1 + f(z))^g(z), where f(z) = 10/z² and g(z) = z².
2. As z approaches infinity, f(z) approaches 0, and g(z) approaches infinity, which suggests the limit is of the form (1 + 0)^∞, an indeterminate form.
3. To resolve this indeterminate form, take the natural logarithm of the expression: ln((1 + 10/z²)ᶻ²) = z² * ln(1 + 10/z²).
4. Now, evaluate the limit of the logarithmic expression: lim_{z→∞} z² * ln(1 + 10/z²). This is another indeterminate form ∞ * 0, which can be rewritten using l'Hôpital's Rule.
5. Rewrite the expression as lim_{z→∞} ln(1 + 10/z²) / (1/z²) and apply l'Hôpital's Rule by differentiating the numerator and the denominator with respect to z, then evaluate the limit.

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 the continuity and differentiability of functions.

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 are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In the context of limits, exponential functions can exhibit unique behaviors as their input approaches infinity or other critical points. Understanding their growth rates is essential for evaluating limits involving expressions raised to a power, especially when combined with polynomial or rational functions.