Question

Find the limit. \(\lim _{x \rightarrow \infty} \tanh x\)

Short answer

The limit of the function \( \tanh x \) as \( x \) approaches infinity is 1.

Step-by-step solution

Understand the behavior of hyperbolic functions

The hyperbolic tangent function is defined as the ratio of the hyperbolic sine function to the hyperbolic cosine function, i.e., \( \tanh x = \frac{\sinh x}{\cosh x} \). As \( x \) becomes larger and larger, both \( \sinh x \) and \( \cosh x \) trends towards infinity. However, the increase in \( \cosh x \) is faster than \( \sinh x \).

Find the limit

As a result, the ratio \( \frac{\sinh x}{\cosh x} \), or \( \tanh x \), approaches 1 as \( x \) approaches infinity. Therefore, \( \lim _{x \rightarrow \infty} \tanh x = 1 \).