Question

Determine the infinite limit.

\(\mathop {\lim }\limits_{x \to {{\left( {\pi /2} \right)}^ + }} \,\frac{1}{x}\sec x\)

Short answer

The limit tends to negative infinity.

Step-by-step solution

Analyze the given function at the given point

The given function has adenominator tending to the value\(\frac{\pi }{2}\)from the right and anumerator tending tonegative infinityas \(x\) approaches just right to \(\frac{\pi }{2}\).

Estimate the limit

Since the denominator tends to a positive value and the numerator of the function tends to negative infinity as\(x\)approaches\(\frac{\pi }{2}\). So,\(\sec x \to - \infty \)as\(x \to \frac{{{\pi ^ + }}}{2}\).

Thefinal limitmust approach tonegative infinityas shown below:

\(\begin{aligned}\mathop {\lim }\limits_{x \to {{\left( {\pi /2} \right)}^ + }} \frac{1}{x}\sec x &= \frac{{ - \infty }}{{{{\left( {\pi /2} \right)}^ + }}}\\ &= - \infty \end{aligned}\)

Thus, the limit tends to negative infinity.