Question

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

Short answer

The limit of the function \(sech(x)\) as x approaches infinity is 0.

Step-by-step solution

Understand the function

The hyperbolic secant function written as sech(x) is the reciprocal of the hyperbolic cosine function. It is equivalent to the expression \(1/ \cosh(x)\). So \(sech(x) = 1/\cosh(x)\). Since the limit involves infinity, it is essential to understand the hyperbolic cosine function behaviour at infinity.

Property of Hyperbolic Cosine function

The hyperbolic cosine function, \(\cosh(x)\), increases exponentially as x approaches infinity. Hence, \(1/\cosh(x)\) tends to zero as x approaches infinity.

Formulate the Limit

From the properties learned in step 2, it can be observed that as x approaches infinity, \(sech(x) = 1/\cosh(x)\) approaches 0. Hence the limit can be written as \(\lim_{x \to \infty} sech(x) = 0\).