Calculus Examples

$lim_{x \to \frac{π}{2}} \frac{1}{x} \sec (x)$

Step 1

Change the two-sided limit into a right sided limit.

$lim_{x \to {(\frac{π}{2})}^{+}} \frac{1}{x} \sec (x)$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$lim_{x \to {(\frac{π}{2})}^{+}} \frac{1}{x} \cdot lim_{x \to {(\frac{π}{2})}^{+}} \sec (x)$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{2}$ .

$\frac{lim_{x \to {(\frac{π}{2})}^{+}} 1}{lim_{x \to {(\frac{π}{2})}^{+}} x} \cdot lim_{x \to {(\frac{π}{2})}^{+}} \sec (x)$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{2}$ .

$\frac{1}{lim_{x \to {(\frac{π}{2})}^{+}} x} \cdot lim_{x \to {(\frac{π}{2})}^{+}} \sec (x)$

Step 3

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

$\frac{1}{\frac{π}{2}} \cdot lim_{x \to {(\frac{π}{2})}^{+}} \sec (x)$

Step 4

As the $x$ values approach $\frac{π}{2}$ from the right, the function values decrease without bound.

$\frac{1}{\frac{π}{2}} \cdot - \infty$

Step 5

Simplify the answer.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{2}{π} \cdot - \infty$

Step 5.2

Multiply $\frac{2}{π}$ by $1$ .

$\frac{2}{π} \cdot - \infty$

Step 5.3

A non-zero constant times infinity is infinity.

$- \infty$