Calculus Examples

$lim_{x \to 90} \frac{\sec (x)}{\cot (x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$lim_{x \to 90} \frac{\sec (x)}{\frac{\cos (x)}{\sin (x)}}$

Step 1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$lim_{x \to 90} \frac{\frac{1}{\cos (x)}}{\frac{\cos (x)}{\sin (x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\cos (x)}{\sin (x)}$ .

$lim_{x \to 90} \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 1.4

Simplify.

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Step 1.4.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$lim_{x \to 90} \sec (x) \frac{\sin (x)}{\cos (x)}$

Step 1.4.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$lim_{x \to 90} \sec (x) \tan (x)$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $90$ .

$lim_{x \to 90} \sec (x) \cdot lim_{x \to 90} \tan (x)$

Step 3

Consider the left sided limit.

$lim_{x \to 90^{-}} \sec (x)$

Step 4

Make a table to show the behavior of the function $\sec (x)$ as $x$ approaches $90$ from the left.

$\begin{array}{c} x & \sec (x) \\ 89.9 & - 2.80438537 \\ 89.99 & - 2.27732646 \\ 89.999 & - 2.23623899 \\ 89.9999 & - 2.23222151 \\ 89.99999 & - 2.23182065 \end{array}$

Step 5

As the $x$ values approach $90$ , the function values approach $- 2.232$ . Thus, the limit of $\sec (x)$ as $x$ approaches $90$ from the left is $- 2.232$ .

$- 2.232$

Step 6

Consider the right sided limit.

$lim_{x \to 90^{+}} \sec (x)$

Step 7

Make a table to show the behavior of the function $\sec (x)$ as $x$ approaches $90$ from the right.

$\begin{array}{c} x & \sec (x) \\ 90.1 & - 1.86885896 \\ 90.01 & - 2.18822675 \\ 90.001 & - 2.22733326 \\ 90.0001 & - 2.23133094 \\ 90.00001 & - 2.2317316 \\ 90.000001 & - 2.23177167 \end{array}$

Step 8

As the $x$ values approach $90$ , the function values approach $- 2.232$ . Thus, the limit of $\sec (x)$ as $x$ approaches $90$ from the right is $- 2.232$ .

$- 2.232 \cdot lim_{x \to 90} \tan (x)$

Step 9

Consider the left sided limit.

$lim_{x \to 90^{-}} \tan (x)$

Step 10

Make a table to show the behavior of the function $\tan (x)$ as $x$ approaches $90$ from the left.

$\begin{array}{c} x & \tan (x) \\ 89.9 & - 2.62003383 \\ 89.99 & - 2.04602439 \\ 89.999 & - 2.00019119 \\ 89.9999 & - 1.99569859 \\ 89.99999 & - 1.99525022 \\ 89.999999 & - 1.99520539 \end{array}$

Step 11

As the $x$ values approach $90$ , the function values approach $- 1.995$ . Thus, the limit of $\tan (x)$ as $x$ approaches $90$ from the left is $- 1.995$ .

$- 1.995$

Step 12

Consider the right sided limit.

$lim_{x \to 90^{+}} \tan (x)$

Step 13

Make a table to show the behavior of the function $\tan (x)$ as $x$ approaches $90$ from the right.

$\begin{array}{c} x & \tan (x) \\ 90.1 & - 1.57880772 \\ 90.01 & - 1.9463649 \\ 90.001 & - 1.9902295 \\ 90.0001 & - 1.99470242 \\ 90.00001 & - 1.9951506 \end{array}$

Step 14

As the $x$ values approach $90$ , the function values approach $- 1.995$ . Thus, the limit of $\tan (x)$ as $x$ approaches $90$ from the right is $- 1.995$ .

$- 2.232 \cdot - 1.995$

Step 15

Multiply $- 2.232$ by $- 1.995$ .

$4.45284$