Calculus Examples

$lim_{x \to 8} \frac{- 4 x^{3} - 4 x^{2} + 2}{5 x^{3} + 3 x^{2} - 2 x}$

Step 1

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

$\frac{lim_{x \to 8} - 4 x^{3} - 4 x^{2} + 2}{lim_{x \to 8} 5 x^{3} + 3 x^{2} - 2 x}$

Step 2

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

$\frac{- lim_{x \to 8} 4 x^{3} - lim_{x \to 8} 4 x^{2} + lim_{x \to 8} 2}{lim_{x \to 8} 5 x^{3} + 3 x^{2} - 2 x}$

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 4 lim_{x \to 8} x^{3} - lim_{x \to 8} 4 x^{2} + lim_{x \to 8} 2}{lim_{x \to 8} 5 x^{3} + 3 x^{2} - 2 x}$

Step 4

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

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - lim_{x \to 8} 4 x^{2} + lim_{x \to 8} 2}{lim_{x \to 8} 5 x^{3} + 3 x^{2} - 2 x}$

Step 5

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 lim_{x \to 8} x^{2} + lim_{x \to 8} 2}{lim_{x \to 8} 5 x^{3} + 3 x^{2} - 2 x}$

Step 6

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

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + lim_{x \to 8} 2}{lim_{x \to 8} 5 x^{3} + 3 x^{2} - 2 x}$

Step 7

Evaluate the limit of $2$ which is constant as $x$ approaches $8$ .

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + 2}{lim_{x \to 8} 5 x^{3} + 3 x^{2} - 2 x}$

Step 8

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

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + 2}{lim_{x \to 8} 5 x^{3} + lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 2 x}$

Step 9

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + 2}{5 lim_{x \to 8} x^{3} + lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 2 x}$

Step 10

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

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + 2}{5 {(lim_{x \to 8} x)}^{3} + lim_{x \to 8} 3 x^{2} - lim_{x \to 8} 2 x}$

Step 11

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

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + 2}{5 {(lim_{x \to 8} x)}^{3} + 3 lim_{x \to 8} x^{2} - lim_{x \to 8} 2 x}$

Step 12

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

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + 2}{5 {(lim_{x \to 8} x)}^{3} + 3 {(lim_{x \to 8} x)}^{2} - lim_{x \to 8} 2 x}$

Step 13

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

$\frac{- 4 {(lim_{x \to 8} x)}^{3} - 4 {(lim_{x \to 8} x)}^{2} + 2}{5 {(lim_{x \to 8} x)}^{3} + 3 {(lim_{x \to 8} x)}^{2} - 2 lim_{x \to 8} x}$

Step 14

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

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Step 14.1