Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2

Combine $x$ and $\frac{2}{2}$ .

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 2

Combine $\frac{x \cdot 2 - π}{2}$ and $\sec (x)$ .

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

Step 3

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

Consider the left sided limit.

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

Step 5

Make a table to show the behavior of the function $(x \cdot 2 - π) \sec (x)$ as $x$ approaches $\frac{π}{2}$ from the left.

$\begin{array}{c} x & (x \cdot 2 - π) \sec (x) \\ 1.47079632 & - 2.00333722 \\ 1.56079632 & - 2.00003333 \\ 1.56979632 & - 2.00000033 \end{array}$

Step 6

As the $x$ values approach $\frac{π}{2}$ , the function values approach $- 2$ . Thus, the limit of $(x \cdot 2 - π) \sec (x)$ as $x$ approaches $\frac{π}{2}$ from the left is $- 2$ .

$- 2$

Step 7

Consider the right sided limit.

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

Step 8

Make a table to show the behavior of the function $(x \cdot 2 - π) \sec (x)$ as $x$ approaches $\frac{π}{2}$ from the right.

$\begin{array}{c} x & (x \cdot 2 - π) \sec (x) \\ 1.67079632 & - 2.00333722 \\ 1.58079632 & - 2.00003333 \\ 1.57179632 & - 2.00000033 \end{array}$

Step 9

As the $x$ values approach $\frac{π}{2}$ , the function values approach $- 2$ . Thus, the limit of $(x \cdot 2 - π) \sec (x)$ as $x$ approaches $\frac{π}{2}$ from the right is $- 2$ .

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Factor $2$ out of $- 2$ .

$\frac{1}{2} \cdot (2 (- 1))$

Step 10.2

Cancel the common factor.

$\frac{1}{2} \cdot (2 \cdot - 1)$

Step 10.3

Rewrite the expression.

$- 1$