Calculus Examples

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

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

Step 1

Split the limit using the Product of Limits Rule on the limit as

x

$x$ approaches

\frac{π}{2}

$\frac{π}{2}$ .

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

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

Step 2

Evaluate the limit of

x

$x$ by plugging in

\frac{π}{2}

$\frac{π}{2}$ for

x

$x$ .

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

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

Step 3

Consider the left sided limit.

lim_{x \to {(\frac{π}{2})}^{-}} \sec^{2} (x)

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

Step 4

As the

x

$x$ values approach

\frac{π}{2}

$\frac{π}{2}$ from the left, the function values increase without bound.

\infty

$\infty$

Step 5

Consider the right sided limit.

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

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

Step 6

As the

x

$x$ values approach

\frac{π}{2}

$\frac{π}{2}$ from the right, the function values increase without bound.

\frac{π}{2} \cdot \infty

$\frac{π}{2} \cdot \infty$

Step 7

A non-zero constant times infinity is infinity.

\infty

$\infty$