Calculus Examples

$lim_{x \to - 8} - 5 x^{3} + 23 x^{2} + 15 x + 9$

Step 1

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

$- lim_{x \to - 8} 5 x^{3} + lim_{x \to - 8} 23 x^{2} + lim_{x \to - 8} 15 x + lim_{x \to - 8} 9$

Step 2

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

$- 5 lim_{x \to - 8} x^{3} + lim_{x \to - 8} 23 x^{2} + lim_{x \to - 8} 15 x + lim_{x \to - 8} 9$

Step 3

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

$- 5 {(lim_{x \to - 8} x)}^{3} + lim_{x \to - 8} 23 x^{2} + lim_{x \to - 8} 15 x + lim_{x \to - 8} 9$

Step 4

Move the term $23$ outside of the limit because it is constant with respect to $x$ .

$- 5 {(lim_{x \to - 8} x)}^{3} + 23 lim_{x \to - 8} x^{2} + lim_{x \to - 8} 15 x + lim_{x \to - 8} 9$

Step 5

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

$- 5 {(lim_{x \to - 8} x)}^{3} + 23 {(lim_{x \to - 8} x)}^{2} + lim_{x \to - 8} 15 x + lim_{x \to - 8} 9$

Step 6

Move the term $15$ outside of the limit because it is constant with respect to $x$ .

$- 5 {(lim_{x \to - 8} x)}^{3} + 23 {(lim_{x \to - 8} x)}^{2} + 15 lim_{x \to - 8} x + lim_{x \to - 8} 9$

Step 7

Evaluate the limit of $9$ which is constant as $x$ approaches $- 8$ .

$- 5 {(lim_{x \to - 8} x)}^{3} + 23 {(lim_{x \to - 8} x)}^{2} + 15 lim_{x \to - 8} x + 9$

Step 8

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

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Step 8.1

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

$- 5 {(- 8)}^{3} + 23 {(lim_{x \to - 8} x)}^{2} + 15 lim_{x \to - 8} x + 9$

Step 8.2

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

$- 5 {(- 8)}^{3} + 23 {(- 8)}^{2} + 15 lim_{x \to - 8} x + 9$

Step 8.3

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

$- 5 {(- 8)}^{3} + 23 {(- 8)}^{2} + 15 \cdot - 8 + 9$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$- 5 \cdot - 512 + 23 {(- 8)}^{2} + 15 \cdot - 8 + 9$

Step 9.1.2

Multiply $- 5$ by $- 512$ .

$2560 + 23 {(- 8)}^{2} + 15 \cdot - 8 + 9$

Step 9.1.3

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

$2560 + 23 \cdot 64 + 15 \cdot - 8 + 9$

Step 9.1.4

Multiply $23$ by $64$ .

$2560 + 1472 + 15 \cdot - 8 + 9$

Step 9.1.5

Multiply $15$ by $- 8$ .

$2560 + 1472 - 120 + 9$

Step 9.2

Add $2560$ and $1472$ .

$4032 - 120 + 9$

Step 9.3

Subtract $120$ from $4032$ .

$3912 + 9$

Step 9.4

Add $3912$ and $9$ .

$3921$