Calculus Examples

$\frac{lim_{x \to 0} \sin (2 x) \cos (x) - \sin (2 x)}{x^{2}}$

Step 1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sin (2 x) \cos (x) - lim_{x \to 0} \sin (2 x)}{x^{2}}$

Step 2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sin (2 x) \cdot lim_{x \to 0} \cos (x) - lim_{x \to 0} \sin (2 x)}{x^{2}}$

Step 3

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

$\frac{\sin (lim_{x \to 0} 2 x) \cdot lim_{x \to 0} \cos (x) - lim_{x \to 0} \sin (2 x)}{x^{2}}$

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (2 lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (x) - lim_{x \to 0} \sin (2 x)}{x^{2}}$

Step 5

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

$\frac{\sin (2 lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) - lim_{x \to 0} \sin (2 x)}{x^{2}}$

Step 6

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

$\frac{\sin (2 lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) - \sin (lim_{x \to 0} 2 x)}{x^{2}}$

Step 7

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (2 lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x) - \sin (2 lim_{x \to 0} x)}{x^{2}}$

Step 8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (2 \cdot 0) \cdot \cos (lim_{x \to 0} x) - \sin (2 lim_{x \to 0} x)}{x^{2}}$

Step 8.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (2 \cdot 0) \cdot \cos (0) - \sin (2 lim_{x \to 0} x)}{x^{2}}$

Step 8.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (2 \cdot 0) \cdot \cos (0) - \sin (2 \cdot 0)}{x^{2}}$

Step 9

Simplify the answer.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Multiply $2$ by $0$ .

$\frac{\sin (0) \cdot \cos (0) - \sin (2 \cdot 0)}{x^{2}}$

Step 9.1.2

The exact value of $\sin (0)$ is $0$ .

$\frac{0 \cdot \cos (0) - \sin (2 \cdot 0)}{x^{2}}$

Step 9.1.3

The exact value of $\cos (0)$ is $1$ .

$\frac{0 \cdot 1 - \sin (2 \cdot 0)}{x^{2}}$

Step 9.1.4

Multiply $0$ by $1$ .

$\frac{0 - \sin (2 \cdot 0)}{x^{2}}$

Step 9.1.5

Multiply $2$ by $0$ .

$\frac{0 - \sin (0)}{x^{2}}$

Step 9.1.6

The exact value of $\sin (0)$ is $0$ .

$\frac{0 - 0}{x^{2}}$

Step 9.1.7

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{x^{2}}$

Step 9.1.8

Add $0$ and $0$ .

$\frac{0}{x^{2}}$

Step 9.2

Divide $0$ by $x^{2}$ .

$0$