Calculus Examples

$lim_{x \to 8} \frac{2 (\cos (7 x - 4 x))}{5 x}$

Step 1

Move the term $\frac{2}{5}$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{5} lim_{x \to 8} \frac{\cos (7 x - 4 x)}{x}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{2}{5} \cdot \frac{lim_{x \to 8} \cos (7 x - 4 x)}{lim_{x \to 8} x}$

Step 3

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

$\frac{2}{5} \cdot \frac{\cos (lim_{x \to 8} 7 x - 4 x)}{lim_{x \to 8} x}$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{2}{5} \cdot \frac{\cos (lim_{x \to 8} 7 x - lim_{x \to 8} 4 x)}{lim_{x \to 8} x}$

Step 5

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

$\frac{2}{5} \cdot \frac{\cos (7 lim_{x \to 8} x - lim_{x \to 8} 4 x)}{lim_{x \to 8} x}$

Step 6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{2}{5} \cdot \frac{\cos (7 lim_{x \to 8} x - 1 \cdot 4 lim_{x \to 8} x)}{lim_{x \to 8} x}$

Step 7

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2}{5} \cdot \frac{\cos (7 \cdot 8 - 1 \cdot 4 lim_{x \to 8} x)}{lim_{x \to 8} x}$

Step 7.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2}{5} \cdot \frac{\cos (7 \cdot 8 - 1 \cdot 4 \cdot 8)}{lim_{x \to 8} x}$

Step 7.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{2}{5} \cdot \frac{\cos (7 \cdot 8 - 1 \cdot 4 \cdot 8)}{8}$

Step 8

Simplify the answer.

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Step 8.1

Cancel the common factor of $2$ .

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Step 8.1.1

Factor $2$ out of $8$ .

$\frac{2}{5} \cdot \frac{\cos (7 \cdot 8 - 1 \cdot 4 \cdot 8)}{2 (4)}$

Step 8.1.2

Cancel the common factor.

$\frac{2}{5} \cdot \frac{\cos (7 \cdot 8 - 1 \cdot 4 \cdot 8)}{2 \cdot 4}$

Step 8.1.3

Rewrite the expression.

$\frac{1}{5} \cdot \frac{\cos (7 \cdot 8 - 1 \cdot 4 \cdot 8)}{4}$

Step 8.2

Multiply $7$ by $8$ .

$\frac{1}{5} \cdot \frac{\cos (56 - 1 \cdot 4 \cdot 8)}{4}$

Step 8.3

Multiply $- 1$ by $4$ .

$\frac{1}{5} \cdot \frac{\cos (56 - 4 \cdot 8)}{4}$

Step 8.4

Multiply $- 4$ by $8$ .

$\frac{1}{5} \cdot \frac{\cos (56 - 32)}{4}$

Step 8.5

Multiply $\frac{1}{5}$ by $\frac{\cos (56 - 32)}{4}$ .

$\frac{\cos (56 - 32)}{5 \cdot 4}$

Step 8.6

Multiply $5$ by $4$ .

$\frac{\cos (56 - 32)}{20}$

Step 8.7

Simplify the numerator.

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Step 8.7.1

Subtract $32$ from $56$ .

$\frac{\cos (24)}{20}$

Step 8.7.2

Evaluate $\cos (24)$ .

$\frac{0.91354545}{20}$

Step 8.8

Divide $0.91354545$ by $20$ .

$0.04567727$