Calculus Examples

$lim_{x \to 8} \cos (\frac{6 x^{4} - 2 x^{3} - 3}{3 x^{4} + 9 x^{2} - 7})$

Step 1

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

$\cos (lim_{x \to 8} \frac{6 x^{4} - 2 x^{3} - 3}{3 x^{4} + 9 x^{2} - 7})$

Step 2

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

$\cos (\frac{lim_{x \to 8} 6 x^{4} - 2 x^{3} - 3}{lim_{x \to 8} 3 x^{4} + 9 x^{2} - 7})$

Step 3

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

$\cos (\frac{lim_{x \to 8} 6 x^{4} - lim_{x \to 8} 2 x^{3} - lim_{x \to 8} 3}{lim_{x \to 8} 3 x^{4} + 9 x^{2} - 7})$

Step 4

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

$\cos (\frac{6 lim_{x \to 8} x^{4} - lim_{x \to 8} 2 x^{3} - lim_{x \to 8} 3}{lim_{x \to 8} 3 x^{4} + 9 x^{2} - 7})$

Step 5

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - lim_{x \to 8} 2 x^{3} - lim_{x \to 8} 3}{lim_{x \to 8} 3 x^{4} + 9 x^{2} - 7})$

Step 6

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

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 lim_{x \to 8} x^{3}) - lim_{x \to 8} 3}{lim_{x \to 8} 3 x^{4} + 9 x^{2} - 7})$

Step 7

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - lim_{x \to 8} 3}{lim_{x \to 8} 3 x^{4} + 9 x^{2} - 7})$

Step 8

Evaluate the limit of $3$ which is constant as $x$ approaches $8$ .

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{lim_{x \to 8} 3 x^{4} + 9 x^{2} - 7})$

Step 9

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

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{lim_{x \to 8} 3 x^{4} + lim_{x \to 8} 9 x^{2} - lim_{x \to 8} 7})$

Step 10

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{3 lim_{x \to 8} x^{4} + lim_{x \to 8} 9 x^{2} - lim_{x \to 8} 7})$

Step 11

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{3 {(lim_{x \to 8} x)}^{4} + lim_{x \to 8} 9 x^{2} - lim_{x \to 8} 7})$

Step 12

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{3 {(lim_{x \to 8} x)}^{4} + 9 lim_{x \to 8} x^{2} - lim_{x \to 8} 7})$

Step 13

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{3 {(lim_{x \to 8} x)}^{4} + 9 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 7})$

Step 14

Evaluate the limit of $7$ which is constant as $x$ approaches $8$ .

$\cos (\frac{6 {(lim_{x \to 8} x)}^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{3 {(lim_{x \to 8} x)}^{4} + 9 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7})$

Step 15

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

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Step 15.1

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

$\cos (\frac{6 \cdot 8^{4} - (2 {(lim_{x \to 8} x)}^{3}) - 1 \cdot 3}{3 {(lim_{x \to 8} x)}^{4} + 9 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7})$

Step 15.2

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

$\cos (\frac{6 \cdot 8^{4} - (2 \cdot 8^{3}) - 1 \cdot 3}{3 {(lim_{x \to 8} x)}^{4} + 9 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7})$

Step 15.3

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

$\cos (\frac{6 \cdot 8^{4} - (2 \cdot 8^{3}) - 1 \cdot 3}{3 \cdot 8^{4} + 9 {(lim_{x \to 8} x)}^{2} - 1 \cdot 7})$

Step 15.4

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

$\cos (\frac{6 \cdot 8^{4} - (2 \cdot 8^{3}) - 1 \cdot 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16

Simplify the answer.

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Step 16.1

Simplify the numerator.

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Step 16.1.1

Raise $8$ to the power of $4$ .

$\cos (\frac{6 \cdot 4096 - 1 \cdot 2 \cdot 8^{3} - 1 \cdot 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.1.2

Multiply $6$ by $4096$ .

$\cos (\frac{24576 - 1 \cdot 2 \cdot 8^{3} - 1 \cdot 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.1.3

Multiply $- 1$ by $2$ .

$\cos (\frac{24576 - 2 \cdot 8^{3} - 1 \cdot 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.1.4

Raise $8$ to the power of $3$ .

$\cos (\frac{24576 - 2 \cdot 512 - 1 \cdot 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.1.5

Multiply $- 2$ by $512$ .

$\cos (\frac{24576 - 1024 - 1 \cdot 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.1.6

Multiply $- 1$ by $3$ .

$\cos (\frac{24576 - 1024 - 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.1.7

Subtract $1024$ from $24576$ .

$\cos (\frac{23552 - 3}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.1.8

Subtract $3$ from $23552$ .

$\cos (\frac{23549}{3 \cdot 8^{4} + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.2

Simplify the denominator.

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Step 16.2.1

Raise $8$ to the power of $4$ .

$\cos (\frac{23549}{3 \cdot 4096 + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.2.2

Multiply $3$ by $4096$ .

$\cos (\frac{23549}{12288 + 9 \cdot 8^{2} - 1 \cdot 7})$

Step 16.2.3

Raise $8$ to the power of $2$ .

$\cos (\frac{23549}{12288 + 9 \cdot 64 - 1 \cdot 7})$

Step 16.2.4

Multiply $9$ by $64$ .

$\cos (\frac{23549}{12288 + 576 - 1 \cdot 7})$

Step 16.2.5

Multiply $- 1$ by $7$ .

$\cos (\frac{23549}{12288 + 576 - 7})$

Step 16.2.6

Add $12288$ and $576$ .

$\cos (\frac{23549}{12864 - 7})$

Step 16.2.7

Subtract $7$ from $12864$ .

$\cos (\frac{23549}{12857})$

Step 16.3

Evaluate $\cos (\frac{23549}{12857})$ .

$- 0.25786605$