Calculus Examples

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

Step 1

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

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

Step 2

Move the term $8$ outside of the limit because it is constant with respect to $z$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

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

$\frac{- 8 \cdot 8^{3} - 5 lim_{z \to 8} z + 2}{z^{3} + 2 z^{2} - 5 z + 3}$

Step 6.2

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

$\frac{- 8 \cdot 8^{3} - 5 \cdot 8 + 2}{z^{3} + 2 z^{2} - 5 z + 3}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$\frac{- 8 \cdot 512 - 5 \cdot 8 + 2}{z^{3} + 2 z^{2} - 5 z + 3}$

Step 7.1.2

Multiply $- 8$ by $512$ .

$\frac{- 4096 - 5 \cdot 8 + 2}{z^{3} + 2 z^{2} - 5 z + 3}$

Step 7.1.3

Multiply $- 5$ by $8$ .

$\frac{- 4096 - 40 + 2}{z^{3} + 2 z^{2} - 5 z + 3}$

Step 7.1.4

Subtract $40$ from $- 4096$ .

$\frac{- 4136 + 2}{z^{3} + 2 z^{2} - 5 z + 3}$

Step 7.1.5

Add $- 4136$ and $2$ .

$\frac{- 4134}{z^{3} + 2 z^{2} - 5 z + 3}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{4134}{z^{3} + 2 z^{2} - 5 z + 3}$