1^∞

Definition

The expression 1 raised to the power of infinity, denoted as 1^∞, is classified as an indeterminate form. This means that its value is not definitively defined and can lead to different outcomes depending on the context of the limits involved. Understanding this term is crucial in calculus, especially when dealing with limits that approach forms that might seem straightforward but require deeper analysis to resolve.

5 Must Know Facts For Your Next Test

1. 1^∞ arises in limits where a base approaches 1 while the exponent approaches infinity, which can result in various outcomes based on the functions involved.
2. It is important to analyze 1^∞ using techniques like L'Hôpital's Rule or algebraic manipulation to determine the actual limit.
3. The outcome of 1^∞ can vary: it may converge to a specific value or diverge based on the behavior of the functions leading to this form.
4. This indeterminate form appears frequently in exponential growth and decay problems, particularly in calculus scenarios involving e and natural logarithms.
5. Recognizing 1^∞ is key to understanding more complex limits, especially those involving exponential functions and their asymptotic behavior.

Review Questions

• How does the indeterminate form 1^∞ differ from determinate forms, and why is it significant in evaluating limits?
  • The indeterminate form 1^∞ differs from determinate forms because it does not provide a clear or fixed value; instead, it indicates that further analysis is required. This significance lies in its occurrence in various limit problems where both the base approaches 1 and the exponent approaches infinity. Understanding this difference helps students recognize when they need to apply additional calculus techniques to find a conclusive result.
• Explain how L'Hôpital's Rule can be applied to resolve the indeterminate form 1^∞, including an example of its use.
  • L'Hôpital's Rule can be applied to resolve the indeterminate form 1^∞ by first transforming the expression into a limit that fits the 0/0 or ∞/∞ forms. For instance, if we have the limit of (e^(x) - 1) as x approaches 0 over x, which results in an indeterminate form, we can differentiate both the numerator and denominator. After applying L'Hôpital's Rule once or multiple times as needed, we can arrive at a determinate value, demonstrating how to handle such forms effectively.
• Analyze a specific example where 1^∞ arises and demonstrate how to evaluate it step by step, showing all necessary calculations.
  • Consider the limit $$ ext{lim}_{x o 0} (1 + x)^{(1/x)}$$ which results in an indeterminate form of 1^∞. To evaluate this, we can rewrite it using logarithms: let y = (1 + x)^(1/x), then take the natural log: ln(y) = (1/x) ln(1 + x). As x approaches 0, both ln(1 + x) approaches 0 and x approaches 0 leading us to a form of 0/0. We apply L'Hôpital's Rule by differentiating ln(1 + x) and x separately, yielding ln(y) approaches 1. Exponentiating gives y approaches e. Thus, $$ ext{lim}_{x o 0} (1 + x)^{(1/x)} = e$$ showing how to handle the indeterminate form effectively.

Related terms

Indeterminate Form: A mathematical expression that does not have a clear or definite value, often encountered in limits and calculus.

Limit: The value that a function approaches as the input approaches some value. Limits are essential for analyzing indeterminate forms.

L'Hôpital's Rule: A method used to evaluate indeterminate forms by differentiating the numerator and denominator until a determinate form is obtained.