Calculus Examples

$lim_{x \to \frac{π}{2}} - \frac{1 + 6 \sec (x)}{1 + 7 \tan (x)}$

Step 1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$- lim_{x \to \frac{π}{2}} \frac{1 + 6 \sec (x)}{1 + 7 \tan (x)}$

Step 2

Consider the left sided limit.

$lim_{x \to {(\frac{π}{2})}^{-}} \frac{1 + 6 \sec (x)}{1 + 7 \tan (x)}$

Step 3

Make a table to show the behavior of the function $\frac{1 + 6 \sec (x)}{1 + 7 \tan (x)}$ as $x$ approaches $\frac{π}{2}$ from the left.

$\begin{array}{c} x & \frac{1 + 6 \sec (x)}{1 + 7 \tan (x)} \\ 1.47079632 & 0.86340439 \\ 1.56079632 & 0.85738945 \\ 1.56979632 & 0.85716369 \end{array}$

Step 4

As the $x$ values approach $\frac{π}{2}$ , the function values approach $0.857$ . Thus, the limit of $\frac{1 + 6 \sec (x)}{1 + 7 \tan (x)}$ as $x$ approaches $\frac{π}{2}$ from the left is $0.857$ .

$0.857$

Step 5

Consider the right sided limit.

$lim_{x \to {(\frac{π}{2})}^{+}} \frac{1 + 6 \sec (x)}{1 + 7 \tan (x)}$

Step 6

Make a table to show the behavior of the function $\frac{1 + 6 \sec (x)}{1 + 7 \tan (x)}$ as $x$ approaches $\frac{π}{2}$ from the right.

$\begin{array}{c} x & \frac{1 + 6 \sec (x)}{1 + 7 \tan (x)} \\ 1.67079632 & 0.85943166 \\ 1.58079632 & 0.85698139 \\ 1.57179632 & 0.85712287 \end{array}$

Step 7

$- 1 \cdot 0.857$

Step 8

Multiply $- 1$ by $0.857$ .

$- 0.857$