Calculus Examples

$lim_{x \to 8} \frac{3 x^{2} - 6 x^{4}}{9 x^{2} - 3 x - 6}$

Step 1

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

$\frac{lim_{x \to 8} 3 x^{2} - 6 x^{4}}{lim_{x \to 8} 9 x^{2} - 3 x - 6}$

Step 2

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

$\frac{lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 6 x^{4}}{lim_{x \to 8} 9 x^{2} - 3 x - 6}$

Step 3

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

$\frac{3 lim_{x \to 8} x^{2} - lim_{x \to 8} 6 x^{4}}{lim_{x \to 8} 9 x^{2} - 3 x - 6}$

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

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 $x$ .

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

Step 12.2

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

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

Step 12.3

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

$\frac{3 \cdot 8^{2} - 6 \cdot 8^{4}}{9 \cdot 8^{2} - 3 lim_{x \to 8} x - 1 \cdot 6}$

Step 12.4

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

$\frac{3 \cdot 8^{2} - 6 \cdot 8^{4}}{9 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}$

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 $2$ .

$\frac{3 \cdot 64 - 6 \cdot 8^{4}}{9 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}$

Step 13.1.2

Multiply $3$ by $64$ .

$\frac{192 - 6 \cdot 8^{4}}{9 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}$

Step 13.1.3

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

$\frac{192 - 6 \cdot 4096}{9 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}$

Step 13.1.4

Multiply $- 6$ by $4096$ .

$\frac{192 - 24576}{9 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}$

Step 13.1.5

Subtract $24576$ from $192$ .

$\frac{- 24384}{9 \cdot 8^{2} - 3 \cdot 8 - 1 \cdot 6}$

Step 13.2

Simplify the denominator.

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Step 13.2.1

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

$\frac{- 24384}{9 \cdot 64 - 3 \cdot 8 - 1 \cdot 6}$

Step 13.2.2

Multiply $9$ by $64$ .

$\frac{- 24384}{576 - 3 \cdot 8 - 1 \cdot 6}$

Step 13.2.3

Multiply $- 3$ by $8$ .

$\frac{- 24384}{576 - 24 - 1 \cdot 6}$

Step 13.2.4

Multiply $- 1$ by $6$ .

$\frac{- 24384}{576 - 24 - 6}$

Step 13.2.5

Subtract $24$ from $576$ .

$\frac{- 24384}{552 - 6}$

Step 13.2.6

Subtract $6$ from $552$ .

$\frac{- 24384}{546}$

Step 13.3

Cancel the common factor of $- 24384$ and $546$ .

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Step 13.3.1

Factor $6$ out of $- 24384$ .

$\frac{6 (- 4064)}{546}$

Step 13.3.2

Cancel the common factors.

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Step 13.3.2.1

Factor $6$ out of $546$ .

$\frac{6 \cdot - 4064}{6 \cdot 91}$

Step 13.3.2.2

Cancel the common factor.

$\frac{6 \cdot - 4064}{6 \cdot 91}$

Step 13.3.2.3

Rewrite the expression.

$\frac{- 4064}{91}$

Step 13.4

Move the negative in front of the fraction.

$- \frac{4064}{91}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$- \frac{4064}{91}$

Decimal Form:

$- 44.65934065 \dots$