Calculus Examples

$lim_{x \to 8} 7 x + 6 - 2 x^{3} \div 3 + 14 x + x^{2}$

Step 1

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

$lim_{x \to 8} 7 x + lim_{x \to 8} 6 - lim_{x \to 8} 2 x^{3} \div 3 + lim_{x \to 8} 14 x + lim_{x \to 8} x^{2}$

Step 2

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

$7 lim_{x \to 8} x + lim_{x \to 8} 6 - lim_{x \to 8} 2 x^{3} \div 3 + lim_{x \to 8} 14 x + lim_{x \to 8} x^{2}$

Step 3

Evaluate the limit of $6$ which is constant as $x$ approaches $8$ .

$7 lim_{x \to 8} x + 6 - lim_{x \to 8} 2 x^{3} \div 3 + lim_{x \to 8} 14 x + lim_{x \to 8} x^{2}$

Step 4

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $x$ .

$7 lim_{x \to 8} x + 6 - \frac{2}{3} lim_{x \to 8} x^{3} + lim_{x \to 8} 14 x + lim_{x \to 8} x^{2}$

Step 5

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

$7 lim_{x \to 8} x + 6 - \frac{2}{3} {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 14 x + lim_{x \to 8} x^{2}$

Step 6

Move the term $14$ outside of the limit because it is constant with respect to $x$ .

$7 lim_{x \to 8} x + 6 - \frac{2}{3} {(lim_{x \to 8} x)}^{3} + 14 lim_{x \to 8} x + lim_{x \to 8} x^{2}$

Step 7

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

$7 lim_{x \to 8} x + 6 - \frac{2}{3} {(lim_{x \to 8} x)}^{3} + 14 lim_{x \to 8} x + {(lim_{x \to 8} x)}^{2}$

Step 8

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

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Step 8.1

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

$7 \cdot 8 + 6 - \frac{2}{3} {(lim_{x \to 8} x)}^{3} + 14 lim_{x \to 8} x + {(lim_{x \to 8} x)}^{2}$

Step 8.2

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

$7 \cdot 8 + 6 - \frac{2}{3} \cdot 8^{3} + 14 lim_{x \to 8} x + {(lim_{x \to 8} x)}^{2}$

Step 8.3

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

$7 \cdot 8 + 6 - \frac{2}{3} \cdot 8^{3} + 14 \cdot 8 + {(lim_{x \to 8} x)}^{2}$

Step 8.4

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

$7 \cdot 8 + 6 - \frac{2}{3} \cdot 8^{3} + 14 \cdot 8 + 8^{2}$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $7$ by $8$ .

$56 + 6 - \frac{2}{3} \cdot 8^{3} + 14 \cdot 8 + 8^{2}$

Step 9.1.2

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

$56 + 6 - \frac{2}{3} \cdot 512 + 14 \cdot 8 + 8^{2}$

Step 9.1.3

Multiply $- \frac{2}{3} \cdot 512$ .

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Step 9.1.3.1

Multiply $512$ by $- 1$ .

$56 + 6 - 512 (\frac{2}{3}) + 14 \cdot 8 + 8^{2}$

Step 9.1.3.2

Combine $- 512$ and $\frac{2}{3}$ .

$56 + 6 + \frac{- 512 \cdot 2}{3} + 14 \cdot 8 + 8^{2}$

Step 9.1.3.3

Multiply $- 512$ by $2$ .

$56 + 6 + \frac{- 1024}{3} + 14 \cdot 8 + 8^{2}$

Step 9.1.4

Move the negative in front of the fraction.

$56 + 6 - \frac{1024}{3} + 14 \cdot 8 + 8^{2}$

Step 9.1.5

Multiply $14$ by $8$ .

$56 + 6 - \frac{1024}{3} + 112 + 8^{2}$

Step 9.1.6

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

$56 + 6 - \frac{1024}{3} + 112 + 64$

Step 9.2

Find the common denominator.

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Step 9.2.1

Write $56$ as a fraction with denominator $1$ .

$\frac{56}{1} + 6 - \frac{1024}{3} + 112 + 64$

Step 9.2.2

Multiply $\frac{56}{1}$ by $\frac{3}{3}$ .

$\frac{56}{1} \cdot \frac{3}{3} + 6 - \frac{1024}{3} + 112 + 64$

Step 9.2.3

Multiply $\frac{56}{1}$ by $\frac{3}{3}$ .

$\frac{56 \cdot 3}{3} + 6 - \frac{1024}{3} + 112 + 64$

Step 9.2.4

Write $6$ as a fraction with denominator $1$ .

$\frac{56 \cdot 3}{3} + \frac{6}{1} - \frac{1024}{3} + 112 + 64$

Step 9.2.5

Multiply $\frac{6}{1}$ by $\frac{3}{3}$ .

$\frac{56 \cdot 3}{3} + \frac{6}{1} \cdot \frac{3}{3} - \frac{1024}{3} + 112 + 64$

Step 9.2.6

Multiply $\frac{6}{1}$ by $\frac{3}{3}$ .

$\frac{56 \cdot 3}{3} + \frac{6 \cdot 3}{3} - \frac{1024}{3} + 112 + 64$

Step 9.2.7

Write $112$ as a fraction with denominator $1$ .

$\frac{56 \cdot 3}{3} + \frac{6 \cdot 3}{3} - \frac{1024}{3} + \frac{112}{1} + 64$

Step 9.2.8

Multiply $\frac{112}{1}$ by $\frac{3}{3}$ .

$\frac{56 \cdot 3}{3} + \frac{6 \cdot 3}{3} - \frac{1024}{3} + \frac{112}{1} \cdot \frac{3}{3} + 64$

Step 9.2.9

Multiply $\frac{112}{1}$ by $\frac{3}{3}$ .

$\frac{56 \cdot 3}{3} + \frac{6 \cdot 3}{3} - \frac{1024}{3} + \frac{112 \cdot 3}{3} + 64$

Step 9.2.10

Write $64$ as a fraction with denominator $1$ .

$\frac{56 \cdot 3}{3} + \frac{6 \cdot 3}{3} - \frac{1024}{3} + \frac{112 \cdot 3}{3} + \frac{64}{1}$

Step 9.2.11

Multiply $\frac{64}{1}$ by $\frac{3}{3}$ .

$\frac{56 \cdot 3}{3} + \frac{6 \cdot 3}{3} - \frac{1024}{3} + \frac{112 \cdot 3}{3} + \frac{64}{1} \cdot \frac{3}{3}$

Step 9.2.12

Multiply $\frac{64}{1}$ by $\frac{3}{3}$ .

$\frac{56 \cdot 3}{3} + \frac{6 \cdot 3}{3} - \frac{1024}{3} + \frac{112 \cdot 3}{3} + \frac{64 \cdot 3}{3}$

Step 9.3

Combine the numerators over the common denominator.

$\frac{56 \cdot 3 + 6 \cdot 3 - 1024 + 112 \cdot 3 + 64 \cdot 3}{3}$

Step 9.4

Simplify each term.

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Step 9.4.1

Multiply $56$ by $3$ .

$\frac{168 + 6 \cdot 3 - 1024 + 112 \cdot 3 + 64 \cdot 3}{3}$

Step 9.4.2

Multiply $6$ by $3$ .

$\frac{168 + 18 - 1024 + 112 \cdot 3 + 64 \cdot 3}{3}$

Step 9.4.3

Multiply $112$ by $3$ .

$\frac{168 + 18 - 1024 + 336 + 64 \cdot 3}{3}$

Step 9.4.4

Multiply $64$ by $3$ .

$\frac{168 + 18 - 1024 + 336 + 192}{3}$

Step 9.5

Add $168$ and $18$ .

$\frac{186 - 1024 + 336 + 192}{3}$

Step 9.6

Subtract $1024$ from $186$ .

$\frac{- 838 + 336 + 192}{3}$

Step 9.7

Add $- 838$ and $336$ .

$\frac{- 502 + 192}{3}$

Step 9.8

Add $- 502$ and $192$ .

$\frac{- 310}{3}$

Step 9.9

Move the negative in front of the fraction.

$- \frac{310}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{310}{3}$

Decimal Form:

$- 103.\overline{3}$