Calculus Examples

$lim_{x \to 1} \frac{\tan (6 x)}{\sin (3 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (6 x)$ in terms of sines and cosines.

$lim_{x \to 1} \frac{\frac{\sin (6 x)}{\cos (6 x)}}{\sin (3 x)}$

Step 1.2

Rewrite $\frac{\frac{\sin (6 x)}{\cos (6 x)}}{\sin (3 x)}$ as a product.

$lim_{x \to 1} \frac{\sin (6 x)}{\cos (6 x)} \cdot \frac{1}{\sin (3 x)}$

Step 1.3

Simplify.

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Step 1.3.1

Convert from $\frac{\sin (6 x)}{\cos (6 x)}$ to $\tan (6 x)$ .

$lim_{x \to 1} \tan (6 x) \frac{1}{\sin (3 x)}$

Step 1.3.2

Convert from $\frac{1}{\sin (3 x)}$ to $\csc (3 x)$ .

$lim_{x \to 1} \tan (6 x) \csc (3 x)$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $1$ .

$lim_{x \to 1} \tan (6 x) \cdot lim_{x \to 1} \csc (3 x)$

Step 3

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

$\tan (lim_{x \to 1} 6 x) \cdot lim_{x \to 1} \csc (3 x)$

Step 4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\tan (6 lim_{x \to 1} x) \cdot lim_{x \to 1} \csc (3 x)$

Step 5

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

$\tan (6 lim_{x \to 1} x) \cdot \csc (lim_{x \to 1} 3 x)$

Step 6

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

$\tan (6 lim_{x \to 1} x) \cdot \csc (3 lim_{x \to 1} x)$

Step 7

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\tan (6 \cdot 1) \cdot \csc (3 lim_{x \to 1} x)$

Step 7.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\tan (6 \cdot 1) \cdot \csc (3 \cdot 1)$

Step 8

Simplify the answer.

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Step 8.1

Multiply $6$ by $1$ .

$\tan (6) \cdot \csc (3 \cdot 1)$

Step 8.2

Evaluate $\tan (6)$ .

$0.10510423 \cdot \csc (3 \cdot 1)$

Step 8.3

Multiply $3$ by $1$ .

$0.10510423 \cdot \csc (3)$

Step 8.4

Evaluate $\csc (3)$ .

$0.10510423 \cdot 19.1073226$

Step 8.5

Multiply $0.10510423$ by $19.1073226$ .

$2.00826053$