Calculus Examples

$lim_{x \to - 3} 5 (3 x^{2} - 24 x + 3)$

Step 1

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

$5 lim_{x \to - 3} 3 x^{2} - 24 x + 3$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 3$ .

$5 (lim_{x \to - 3} 3 x^{2} - lim_{x \to - 3} 24 x + lim_{x \to - 3} 3)$

Step 3

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

$5 (3 lim_{x \to - 3} x^{2} - lim_{x \to - 3} 24 x + lim_{x \to - 3} 3)$

Step 4

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

$5 (3 {(lim_{x \to - 3} x)}^{2} - lim_{x \to - 3} 24 x + lim_{x \to - 3} 3)$

Step 5

Move the term $24$ outside of the limit because it is constant with respect to $x$ .

$5 (3 {(lim_{x \to - 3} x)}^{2} - 24 lim_{x \to - 3} x + lim_{x \to - 3} 3)$

Step 6

Evaluate the limit of $3$ which is constant as $x$ approaches $- 3$ .

$5 (3 {(lim_{x \to - 3} x)}^{2} - 24 lim_{x \to - 3} x + 3)$

Step 7

Evaluate the limits by plugging in $- 3$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$5 (3 {(- 3)}^{2} - 24 lim_{x \to - 3} x + 3)$

Step 7.2

Evaluate the limit of $x$ by plugging in $- 3$ for $x$ .

$5 (3 {(- 3)}^{2} - 24 \cdot - 3 + 3)$

Step 8

Simplify the answer.

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Step 8.1

Simplify each term.

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Step 8.1.1

Raise $- 3$ to the power of $2$ .

$5 (3 \cdot 9 - 24 \cdot - 3 + 3)$

Step 8.1.2

Multiply $3$ by $9$ .

$5 (27 - 24 \cdot - 3 + 3)$

Step 8.1.3

Multiply $- 24$ by $- 3$ .

$5 (27 + 72 + 3)$

Step 8.2

Add $27$ and $72$ .

$5 (99 + 3)$

Step 8.3

Add $99$ and $3$ .

$5 \cdot 102$

Step 8.4

Multiply $5$ by $102$ .

$510$