Limit Evaluate lim x → ∞ (tanh x)ˣ.

Verified step by step guidance

1. Recall the definition of the hyperbolic tangent function: \(\tanh x = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\).
2. Analyze the behavior of \(\tanh x\) as \(x \to \infty\). Since \(e^{x}\) grows much faster than \(e^{-x}\), \(\tanh x\) approaches 1 from below.
3. Express the limit in the form \(\lim_{x \to \infty} (\tanh x)^{x} = \lim_{x \to \infty} \left(1 - \epsilon_x\right)^{x}\), where \(\epsilon_x\) is a small positive number approaching 0 as \(x \to \infty\).
4. Use the approximation \(\ln(1 - \epsilon_x) \approx -\epsilon_x\) for small \(\epsilon_x\) to rewrite the limit as \(\lim_{x \to \infty} e^{x \ln(\tanh x)} \approx \lim_{x \to \infty} e^{-x \epsilon_x}\).
5. Determine the rate at which \(\epsilon_x\) approaches 0 to evaluate the limit of the exponent \(-x \epsilon_x\) and thus find the behavior of the original limit.

Key Concepts

Behavior of the Hyperbolic Tangent Function (tanh x) as x → ∞

The hyperbolic tangent function, tanh x, approaches 1 as x approaches infinity. Understanding this limit is crucial because it determines the base of the expression (tanh x)^x for large x values.

Limits Involving Indeterminate Forms

Expressions like (tanh x)^x can lead to indeterminate forms such as 1^∞. Recognizing and resolving these forms often requires rewriting the expression using logarithms or other techniques to evaluate the limit accurately.

Use of Logarithms to Evaluate Limits of Exponential Forms

Taking the natural logarithm of an expression like (tanh x)^x transforms it into x * ln(tanh x), simplifying the limit evaluation. After finding the limit of the logarithm, exponentiate the result to obtain the original limit.