Calculus Examples

$lim_{x \to 8} \frac{3 x^{2} + 20 x}{2 x^{4} \cdot (3 x^{3}) - 29}$

Step 1

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

$\frac{lim_{x \to 8} 3 x^{2} + 20 x}{lim_{x \to 8} 2 x^{4} \cdot (3 x^{3}) - 29}$

Step 2

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

$\frac{lim_{x \to 8} 3 x^{2} + lim_{x \to 8} 20 x}{lim_{x \to 8} 2 x^{4} \cdot (3 x^{3}) - 29}$

Step 3

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

$\frac{3 lim_{x \to 8} x^{2} + lim_{x \to 8} 20 x}{lim_{x \to 8} 2 x^{4} \cdot (3 x^{3}) - 29}$

Step 4

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

$\frac{3 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 20 x}{lim_{x \to 8} 2 x^{4} \cdot (3 x^{3}) - 29}$

Step 5

Move the term $20$ outside of the limit because it is constant with respect to $x$ .

$\frac{3 {(lim_{x \to 8} x)}^{2} + 20 lim_{x \to 8} x}{lim_{x \to 8} 2 x^{4} \cdot (3 x^{3}) - 29}$

Step 6

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

$\frac{3 {(lim_{x \to 8} x)}^{2} + 20 lim_{x \to 8} x}{lim_{x \to 8} 2 x^{4} \cdot 3 x^{3} - lim_{x \to 8} 29}$

Step 7

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

$\frac{3 {(lim_{x \to 8} x)}^{2} + 20 lim_{x \to 8} x}{2 \cdot 3 lim_{x \to 8} x^{4} x^{3} - lim_{x \to 8} 29}$

Step 8

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

$\frac{3 {(lim_{x \to 8} x)}^{2} + 20 lim_{x \to 8} x}{2 \cdot 3 lim_{x \to 8} x^{4} \cdot lim_{x \to 8} x^{3} - lim_{x \to 8} 29}$

Step 9

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

$\frac{3 {(lim_{x \to 8} x)}^{2} + 20 lim_{x \to 8} x}{2 \cdot 3 {(lim_{x \to 8} x)}^{4} \cdot lim_{x \to 8} x^{3} - lim_{x \to 8} 29}$

Step 10

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

$\frac{3 {(lim_{x \to 8} x)}^{2} + 20 lim_{x \to 8} x}{2 \cdot 3 {(lim_{x \to 8} x)}^{4} \cdot {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 29}$

Step 11

Evaluate the limit of $29$ which is constant as $x$ approaches $8$ .

$\frac{3 {(lim_{x \to 8} x)}^{2} + 20 lim_{x \to 8} x}{2 \cdot 3 {(lim_{x \to 8} x)}^{4} \cdot {(lim_{x \to 8} x)}^{3} - 1 \cdot 29}$

Step 12

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

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Step 12.1

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

$\frac{3 \cdot 8^{2} + 20 lim_{x \to 8} x}{2 \cdot 3 {(lim_{x \to 8} x)}^{4} \cdot {(lim_{x \to 8} x)}^{3} - 1 \cdot 29}$

Step 12.2

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

$\frac{3 \cdot 8^{2} + 20 \cdot 8}{2 \cdot 3 {(lim_{x \to 8} x)}^{4} \cdot {(lim_{x \to 8} x)}^{3} - 1 \cdot 29}$

Step 12.3

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

$\frac{3 \cdot 8^{2} + 20 \cdot 8}{2 \cdot 3 \cdot 8^{4} \cdot {(lim_{x \to 8} x)}^{3} - 1 \cdot 29}$

Step 12.4

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

$\frac{3 \cdot 8^{2} + 20 \cdot 8}{2 \cdot 3 \cdot 8^{4} \cdot 8^{3} - 1 \cdot 29}$

Step 13

Simplify the answer.

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Step 13.1

Simplify the numerator.

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Step 13.1.1

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

$\frac{3 \cdot 64 + 20 \cdot 8}{2 \cdot 3 \cdot 8^{4} \cdot 8^{3} - 1 \cdot 29}$

Step 13.1.2

Multiply $3$ by $64$ .

$\frac{192 + 20 \cdot 8}{2 \cdot 3 \cdot 8^{4} \cdot 8^{3} - 1 \cdot 29}$

Step 13.1.3

Multiply $20$ by $8$ .

$\frac{192 + 160}{2 \cdot 3 \cdot 8^{4} \cdot 8^{3} - 1 \cdot 29}$

Step 13.1.4

Add $192$ and $160$ .

$\frac{352}{2 \cdot 3 \cdot 8^{4} \cdot 8^{3} - 1 \cdot 29}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

Multiply $8^{4}$ by $8^{3}$ by adding the exponents.

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Step 13.2.1.1

Move $8^{3}$ .

$\frac{352}{2 \cdot 3 \cdot (8^{3} \cdot 8^{4}) - 1 \cdot 29}$

Step 13.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{352}{2 \cdot 3 \cdot 8^{3 + 4} - 1 \cdot 29}$

Step 13.2.1.3

Add $3$ and $4$ .

$\frac{352}{2 \cdot 3 \cdot 8^{7} - 1 \cdot 29}$

Step 13.2.2

Multiply $2$ by $3$ .

$\frac{352}{6 \cdot 8^{7} - 1 \cdot 29}$

Step 13.2.3

Raise $8$ to the power of $7$ .

$\frac{352}{6 \cdot 2097152 - 1 \cdot 29}$

Step 13.2.4

Multiply $6$ by $2097152$ .

$\frac{352}{12582912 - 1 \cdot 29}$

Step 13.2.5

Multiply $- 1$ by $29$ .

$\frac{352}{12582912 - 29}$

Step 13.2.6

Subtract $29$ from $12582912$ .

$\frac{352}{12582883}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$\frac{352}{12582883}$

Decimal Form:

$2.7974511 \cdot 10^{- 5}$