Calculus Examples

$\frac{lim_{x \to 0} 7 \sin (x) \cos (x)}{5 x}$

Step 1

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

$\frac{7 lim_{x \to 0} \sin (x) \cos (x)}{5 x}$

Step 2

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

$\frac{7 lim_{x \to 0} \sin (x) \cdot lim_{x \to 0} \cos (x)}{5 x}$

Step 3

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

$\frac{7 \sin (lim_{x \to 0} x) \cdot lim_{x \to 0} \cos (x)}{5 x}$

Step 4

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

$\frac{7 \sin (lim_{x \to 0} x) \cdot \cos (lim_{x \to 0} x)}{5 x}$

Step 5

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

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Step 5.1

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

$\frac{7 \sin (0) \cdot \cos (lim_{x \to 0} x)}{5 x}$

Step 5.2

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

$\frac{7 \sin (0) \cdot \cos (0)}{5 x}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$\frac{7 \cdot 0 \cos (0)}{5 x}$

Step 6.1.2

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

$\frac{7 \cdot 0 \cdot 1}{5 x}$

Step 6.1.3

Combine exponents.

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Step 6.1.3.1

Multiply $7$ by $0$ .

$\frac{0 \cdot 1}{5 x}$

Step 6.1.3.2

Multiply $0$ by $1$ .

$\frac{0}{5 x}$

Step 6.2

Cancel the common factor of $0$ and $5$ .

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Step 6.2.1

Factor $5$ out of $0$ .

$\frac{5 (0)}{5 x}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $5$ out of $5 x$ .

$\frac{5 (0)}{5 (x)}$

Step 6.2.2.2

Cancel the common factor.

$\frac{5 \cdot 0}{5 x}$

Step 6.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 6.3

Divide $0$ by $x$ .

$0$