Calculus Examples

$lim_{x \to 1} \frac{\sin (8 x)}{\tan (7 x)}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (7 x)$ in terms of sines and cosines.

$lim_{x \to 1} \frac{\sin (8 x)}{\frac{\sin (7 x)}{\cos (7 x)}}$

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (7 x)}{\cos (7 x)}$ .

$lim_{x \to 1} \frac{\sin (8 x) \cos (7 x)}{\sin (7 x)}$

Step 1.3

Convert from $\frac{\sin (8 x) \cos (7 x)}{\sin (7 x)}$ to $\sin (8 x) \cot (7 x)$ .

$lim_{x \to 1} \sin (8 x) \cot (7 x)$

Step 2

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

$lim_{x \to 1} \sin (8 x) \cdot lim_{x \to 1} \cot (7 x)$

Step 3

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

$\sin (lim_{x \to 1} 8 x) \cdot lim_{x \to 1} \cot (7 x)$

Step 4

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\sin (8 lim_{x \to 1} x) \cdot lim_{x \to 1} \cot (7 x)$

Step 5

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

$\sin (8 lim_{x \to 1} x) \cdot \cot (lim_{x \to 1} 7 x)$

Step 6

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\sin (8 lim_{x \to 1} x) \cdot \cot (7 lim_{x \to 1} x)$

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 $x$ .

$\sin (8 \cdot 1) \cdot \cot (7 lim_{x \to 1} x)$

Step 7.2

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

$\sin (8 \cdot 1) \cdot \cot (7 \cdot 1)$

Step 8

Simplify the answer.

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Step 8.1

Multiply $8$ by $1$ .

$\sin (8) \cdot \cot (7 \cdot 1)$

Step 8.2

Evaluate $\sin (8)$ .

$0.1391731 \cdot \cot (7 \cdot 1)$

Step 8.3

Multiply $7$ by $1$ .

$0.1391731 \cdot \cot (7)$

Step 8.4

Evaluate $\cot (7)$ .

$0.1391731 \cdot 8.14434642$

Step 8.5

Multiply $0.1391731$ by $8.14434642$ .

$1.13347394$