Calculus Examples

$lim_{x \to 90} \frac{\tan (x)}{\tan (3 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (3 x)$ in terms of sines and cosines.

$lim_{x \to 90} \frac{\tan (x)}{\frac{\sin (3 x)}{\cos (3 x)}}$

Step 1.2

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

$lim_{x \to 90} \frac{\frac{\sin (x)}{\cos (x)}}{\frac{\sin (3 x)}{\cos (3 x)}}$

Step 1.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (3 x)}{\cos (3 x)}$ .

$lim_{x \to 90} \frac{\sin (x)}{\cos (x)} \cdot \frac{\cos (3 x)}{\sin (3 x)}$

Step 1.4

Simplify.

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Step 1.4.1

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

$lim_{x \to 90} \tan (x) \frac{\cos (3 x)}{\sin (3 x)}$

Step 1.4.2

Convert from $\frac{\cos (3 x)}{\sin (3 x)}$ to $\cot (3 x)$ .

$lim_{x \to 90} \tan (x) \cot (3 x)$

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 $90$ .

$lim_{x \to 90} \tan (x) \cdot lim_{x \to 90} \cot (3 x)$

Step 2.2

Move the limit inside the trig function because cotangent is continuous.

$lim_{x \to 90} \tan (x) \cdot \cot (lim_{x \to 90} 3 x)$

Step 2.3

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$lim_{x \to 90} \tan (x) \cdot \cot (3 lim_{x \to 90} x)$

Step 3

Evaluate the limit of $x$ by plugging in $90$ for $x$ .

$lim_{x \to 90} \tan (x) \cdot \cot (3 \cdot 90)$

Step 4

Consider the left sided limit.

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

Step 5

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 6

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 7

Consider the right sided limit.

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

Step 8

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 9

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 $- 1.995$ .

$- 1.995 \cdot \cot (3 \cdot 90)$

Step 10

Simplify the answer.

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Step 10.1

Multiply $3$ by $90$ .

$- 1.995 \cdot \cot (270)$

Step 10.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 1.995 \cdot \cot (90)$

Step 10.3

The exact value of $\cot (90)$ is $0$ .

$- 1.995 \cdot 0$

Step 10.4

Multiply $- 1.995$ by $0$ .

$0$