Calculus Examples

$lim_{x \to 3} 2 x^{3} - 12 x^{2} + 5 x + 3$

Step 1

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

$lim_{x \to 3} 2 x^{3} - lim_{x \to 3} 12 x^{2} + lim_{x \to 3} 5 x + lim_{x \to 3} 3$

Step 2

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

$2 lim_{x \to 3} x^{3} - lim_{x \to 3} 12 x^{2} + lim_{x \to 3} 5 x + lim_{x \to 3} 3$

Step 3

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

$2 {(lim_{x \to 3} x)}^{3} - lim_{x \to 3} 12 x^{2} + lim_{x \to 3} 5 x + lim_{x \to 3} 3$

Step 4

Move the term $12$ outside of the limit because it is constant with respect to $x$ .

$2 {(lim_{x \to 3} x)}^{3} - 12 lim_{x \to 3} x^{2} + lim_{x \to 3} 5 x + lim_{x \to 3} 3$

Step 5

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

$2 {(lim_{x \to 3} x)}^{3} - 12 {(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 5 x + lim_{x \to 3} 3$

Step 6

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

$2 {(lim_{x \to 3} x)}^{3} - 12 {(lim_{x \to 3} x)}^{2} + 5 lim_{x \to 3} x + lim_{x \to 3} 3$

Step 7

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

$2 {(lim_{x \to 3} x)}^{3} - 12 {(lim_{x \to 3} x)}^{2} + 5 lim_{x \to 3} x + 3$

Step 8

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

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Step 8.1

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

$2 \cdot 3^{3} - 12 {(lim_{x \to 3} x)}^{2} + 5 lim_{x \to 3} x + 3$

Step 8.2

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

$2 \cdot 3^{3} - 12 \cdot 3^{2} + 5 lim_{x \to 3} x + 3$

Step 8.3

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

$2 \cdot 3^{3} - 12 \cdot 3^{2} + 5 \cdot 3 + 3$

Step 9

Simplify the answer.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$2 \cdot 27 - 12 \cdot 3^{2} + 5 \cdot 3 + 3$

Step 9.1.2

Multiply $2$ by $27$ .

$54 - 12 \cdot 3^{2} + 5 \cdot 3 + 3$

Step 9.1.3

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

$54 - 12 \cdot 9 + 5 \cdot 3 + 3$

Step 9.1.4

Multiply $- 12$ by $9$ .

$54 - 108 + 5 \cdot 3 + 3$

Step 9.1.5

Multiply $5$ by $3$ .

$54 - 108 + 15 + 3$

Step 9.2

Subtract $108$ from $54$ .

$- 54 + 15 + 3$

Step 9.3

Add $- 54$ and $15$ .

$- 39 + 3$

Step 9.4

Add $- 39$ and $3$ .

$- 36$