Calculus Examples

$\frac{lim_{x \to 1} (x) (\sin (x))}{\cos (3 x)} + \frac{\sin (3 x)}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 1

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

$\frac{lim_{x \to 1} x \cdot lim_{x \to 1} \sin (x)}{\cos (3 x)} + \frac{\sin (3 x)}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 2

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

$\frac{lim_{x \to 1} x \cdot \sin (lim_{x \to 1} x)}{\cos (3 x)} + \frac{\sin (3 x)}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 3

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

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Step 3.1

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

$\frac{1 \cdot \sin (lim_{x \to 1} x)}{\cos (3 x)} + \frac{\sin (3 x)}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 3.2

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

$\frac{1 \cdot \sin (1)}{\cos (3 x)} + \frac{\sin (3 x)}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 4

Simplify each term.

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Step 4.1

Convert from $\frac{1 \cdot \sin (1)}{\cos (3 x)}$ to $1 \tan (3 x)$ .

$1 \tan (3 x) + \frac{\sin (3 x)}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 4.2

Multiply $\tan (3 x)$ by $1$ .

$\tan (3 x) + \frac{\sin (3 x)}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 4.3

Rewrite $\frac{\sin (3 x)}{\cos (9 x)}$ as a product.

$\tan (3 x) + \sin (3 x) \frac{1}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 4.4

Write $\sin (3 x)$ as a fraction with denominator $1$ .

$\tan (3 x) + \frac{\sin (3 x)}{1} \cdot \frac{1}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 4.5

Simplify.

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Step 4.5.1

Divide $\sin (3 x)$ by $1$ .

$\tan (3 x) + \sin (3 x) \frac{1}{\cos (9 x)} + \frac{\sin (9 x)}{\cos (27 x)}$

Step 4.5.2

Convert from $\frac{1}{\cos (9 x)}$ to $\sec (9 x)$ .

$\tan (3 x) + \sin (3 x) \sec (9 x) + \frac{\sin (9 x)}{\cos (27 x)}$

Step 4.6

Rewrite $\frac{\sin (9 x)}{\cos (27 x)}$ as a product.

$\tan (3 x) + \sin (3 x) \sec (9 x) + \sin (9 x) \frac{1}{\cos (27 x)}$

Step 4.7

Write $\sin (9 x)$ as a fraction with denominator $1$ .

$\tan (3 x) + \sin (3 x) \sec (9 x) + \frac{\sin (9 x)}{1} \cdot \frac{1}{\cos (27 x)}$

Step 4.8

Simplify.

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Step 4.8.1

Divide $\sin (9 x)$ by $1$ .

$\tan (3 x) + \sin (3 x) \sec (9 x) + \sin (9 x) \frac{1}{\cos (27 x)}$

Step 4.8.2

Convert from $\frac{1}{\cos (27 x)}$ to $\sec (27 x)$ .

$\tan (3 x) + \sin (3 x) \sec (9 x) + \sin (9 x) \sec (27 x)$