Calculus Examples

$lim_{x \to 8} 6 \cos (\frac{6}{7 x})$

Step 1

Evaluate the limit.

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Step 1.1

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$6 lim_{x \to 8} \cos (\frac{6}{7 x})$

Step 1.2

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

$6 \cos (lim_{x \to 8} \frac{6}{7 x})$

Step 1.3

Move the term $\frac{6}{7}$ outside of the limit because it is constant with respect to $x$ .

$6 \cos (\frac{6}{7} lim_{x \to 8} \frac{1}{x})$

Step 1.4

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

$6 \cos (\frac{6}{7} \cdot \frac{lim_{x \to 8} 1}{lim_{x \to 8} x})$

Step 1.5

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

$6 \cos (\frac{6}{7} \cdot \frac{1}{lim_{x \to 8} x})$

Step 2

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

$6 \cos (\frac{6}{7} \cdot \frac{1}{8})$

Step 3

Simplify the answer.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Factor $2$ out of $6$ .

$6 \cos (\frac{2 (3)}{7} \cdot \frac{1}{8})$

Step 3.1.2

Factor $2$ out of $8$ .

$6 \cos (\frac{2 \cdot 3}{7} \cdot \frac{1}{2 \cdot 4})$

Step 3.1.3

Cancel the common factor.

$6 \cos (\frac{2 \cdot 3}{7} \cdot \frac{1}{2 \cdot 4})$

Step 3.1.4

Rewrite the expression.

$6 \cos (\frac{3}{7} \cdot \frac{1}{4})$

Step 3.2

Multiply $\frac{3}{7}$ by $\frac{1}{4}$ .

$6 \cos (\frac{3}{7 \cdot 4})$

Step 3.3

Multiply $7$ by $4$ .

$6 \cos (\frac{3}{28})$

Step 3.4

Evaluate $\cos (\frac{3}{28})$ .

$6 \cdot 0.99426569$

Step 3.5

Multiply $6$ by $0.99426569$ .

$5.96559415$