Calculus Examples

$lim_{x \to 8} (9261 - 9261) (\frac{1}{3} + \frac{1}{2 x} + \frac{1}{6 x^{2}})$

Step 1

Move the term $9261 - 9261$ outside of the limit because it is constant with respect to $x$ .

$(9261 - 9261) lim_{x \to 8} \frac{1}{3} + \frac{1}{2 x} + \frac{1}{6 x^{2}}$

Step 2

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

$(9261 - 9261) (lim_{x \to 8} \frac{1}{3} + lim_{x \to 8} \frac{1}{2 x} + lim_{x \to 8} \frac{1}{6 x^{2}})$

Step 3

Evaluate the limit of $\frac{1}{3}$ which is constant as $x$ approaches $8$ .

$(9261 - 9261) (\frac{1}{3} + lim_{x \to 8} \frac{1}{2 x} + lim_{x \to 8} \frac{1}{6 x^{2}})$

Step 4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} lim_{x \to 8} \frac{1}{x} + lim_{x \to 8} \frac{1}{6 x^{2}})$

Step 5

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

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{lim_{x \to 8} 1}{lim_{x \to 8} x} + lim_{x \to 8} \frac{1}{6 x^{2}})$

Step 6

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

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{lim_{x \to 8} x} + lim_{x \to 8} \frac{1}{6 x^{2}})$

Step 7

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $x$ .

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{lim_{x \to 8} x} + \frac{1}{6} lim_{x \to 8} \frac{1}{x^{2}})$

Step 8

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

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{lim_{x \to 8} x} + \frac{1}{6} \cdot \frac{lim_{x \to 8} 1}{lim_{x \to 8} x^{2}})$

Step 9

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

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{lim_{x \to 8} x} + \frac{1}{6} \cdot \frac{1}{lim_{x \to 8} x^{2}})$

Step 10

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

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{lim_{x \to 8} x} + \frac{1}{6} \cdot \frac{1}{{(lim_{x \to 8} x)}^{2}})$

Step 11

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

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Step 11.1

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

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{8} + \frac{1}{6} \cdot \frac{1}{{(lim_{x \to 8} x)}^{2}})$

Step 11.2

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

$(9261 - 9261) (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{8} + \frac{1}{6} \cdot \frac{1}{8^{2}})$

Step 12

Simplify the answer.

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Step 12.1

Subtract $9261$ from $9261$ .

$0 (\frac{1}{3} + \frac{1}{2} \cdot \frac{1}{8} + \frac{1}{6} \cdot \frac{1}{8^{2}})$

Step 12.2

Simplify each term.

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Step 12.2.1

Multiply $\frac{1}{2} \cdot \frac{1}{8}$ .

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Step 12.2.1.1

Multiply $\frac{1}{2}$ by $\frac{1}{8}$ .

$0 (\frac{1}{3} + \frac{1}{2 \cdot 8} + \frac{1}{6} \cdot \frac{1}{8^{2}})$

Step 12.2.1.2

Multiply $2$ by $8$ .

$0 (\frac{1}{3} + \frac{1}{16} + \frac{1}{6} \cdot \frac{1}{8^{2}})$

Step 12.2.2

Combine.

$0 (\frac{1}{3} + \frac{1}{16} + \frac{1 \cdot 1}{6 \cdot 8^{2}})$

Step 12.2.3

Multiply $1$ by $1$ .

$0 (\frac{1}{3} + \frac{1}{16} + \frac{1}{6 \cdot 8^{2}})$

Step 12.2.4

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

$0 (\frac{1}{3} + \frac{1}{16} + \frac{1}{6 \cdot 64})$

Step 12.2.5

Multiply $6$ by $64$ .

$0 (\frac{1}{3} + \frac{1}{16} + \frac{1}{384})$

Step 12.3

Find the common denominator.

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Step 12.3.1

Multiply $\frac{1}{3}$ by $\frac{128}{128}$ .

$0 (\frac{1}{3} \cdot \frac{128}{128} + \frac{1}{16} + \frac{1}{384})$

Step 12.3.2

Multiply $\frac{1}{3}$ by $\frac{128}{128}$ .

$0 (\frac{128}{3 \cdot 128} + \frac{1}{16} + \frac{1}{384})$

Step 12.3.3

Multiply $\frac{1}{16}$ by $\frac{24}{24}$ .

$0 (\frac{128}{3 \cdot 128} + \frac{1}{16} \cdot \frac{24}{24} + \frac{1}{384})$

Step 12.3.4

Multiply $\frac{1}{16}$ by $\frac{24}{24}$ .

$0 (\frac{128}{3 \cdot 128} + \frac{24}{16 \cdot 24} + \frac{1}{384})$

Step 12.3.5

Multiply $3$ by $128$ .

$0 (\frac{128}{384} + \frac{24}{16 \cdot 24} + \frac{1}{384})$

Step 12.3.6

Multiply $16$ by $24$ .

$0 (\frac{128}{384} + \frac{24}{384} + \frac{1}{384})$

Step 12.4

Combine the numerators over the common denominator.

$0 \frac{128 + 24 + 1}{384}$

Step 12.5

Add $128$ and $24$ .

$0 \frac{152 + 1}{384}$

Step 12.6

Add $152$ and $1$ .

$0 (\frac{153}{384})$

Step 12.7

Cancel the common factor of $153$ and $384$ .

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Step 12.7.1

Factor $3$ out of $153$ .

$0 \frac{3 (51)}{384}$

Step 12.7.2

Cancel the common factors.

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Step 12.7.2.1

Factor $3$ out of $384$ .

$0 \frac{3 \cdot 51}{3 \cdot 128}$

Step 12.7.2.2

Cancel the common factor.

$0 \frac{3 \cdot 51}{3 \cdot 128}$

Step 12.7.2.3

Rewrite the expression.

$0 (\frac{51}{128})$

Step 12.8

Multiply $0$ by $\frac{51}{128}$ .

$0$