Question

Find the limit or show that it does not exist.

39. \(\mathop {{\rm{lim}}}\limits_{x \to {{\left( {\pi /2} \right)}^ + }} {e^{\sec x}}\)

Short answer

The value of the limit is \(\mathop {{\rm{lim}}}\limits_{x \to {{\left( {\pi /2} \right)}^ + }} {e^{{\rm{sec}}x}} = 0\).

Step-by-step solution

Determine the nature of \({\rm{sec}}x\)

The value of \({\rm{sec}}x\) tends to \( - \infty \) as the value of \(x\) tends to \(\pi /2\) from the right.

Find the limit

From step 1, this implies that \({e^{{\rm{sec}}x}}\) tends to \(0\) as the value of \(x\) tends to \(\pi /2\) from the right, that is, \(\mathop {{\rm{lim}}}\limits_{x \to {{\left( {\pi /2} \right)}^ + }} {e^{{\rm{sec}}x}} = 0\).

Thus, the value of the limit is \(0\).