88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.


lim x → ∞ (1 − coth x) / (1 − tanh x)

Verified step by step guidance

1. Identify the limit expression: \(\lim_{x \to \infty} \frac{1 - \coth x}{1 - \tanh x}\).
2. Recall the definitions of hyperbolic functions: \(\coth x = \frac{\cosh x}{\sinh x}\) and \(\tanh x = \frac{\sinh x}{\cosh x}\).
3. Evaluate the behavior of \(\coth x\) and \(\tanh x\) as \(x \to \infty\) to check if the limit is an indeterminate form suitable for l'Hôpital's Rule.
4. If the limit is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), apply l'Hôpital's Rule by differentiating numerator and denominator separately with respect to \(x\):
5. Compute \(\frac{d}{dx} (1 - \coth x)\) and \(\frac{d}{dx} (1 - \tanh x)\), then form the new limit \(\lim_{x \to \infty} \frac{\frac{d}{dx} (1 - \coth x)}{\frac{d}{dx} (1 - \tanh x)}\) and analyze this limit.

Key Concepts

Hyperbolic Functions (coth and tanh)

Hyperbolic functions like coth(x) and tanh(x) are analogs of trigonometric functions but based on hyperbolas. As x approaches infinity, tanh(x) approaches 1, while coth(x) also approaches 1, but understanding their behavior is crucial for evaluating limits involving these functions.

Limits at Infinity

Evaluating limits as x approaches infinity involves analyzing the behavior of functions for very large values of x. Recognizing how functions like coth(x) and tanh(x) behave at infinity helps determine the form of the limit and whether it is indeterminate.

l’Hôpital’s Rule

l’Hôpital’s Rule is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator separately. Applying this rule simplifies complex limits, especially when direct substitution leads to indeterminate expressions.