Calculus Examples

$lim_{x \to 2} (x + 1) (3 x^{2} - 9)$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $2$ .

$lim_{x \to 2} x + 1 \cdot lim_{x \to 2} 3 x^{2} - 9$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$(lim_{x \to 2} x + lim_{x \to 2} 1) \cdot lim_{x \to 2} 3 x^{2} - 9$

Step 3

Evaluate the limit of $1$ which is constant as $x$ approaches $2$ .

$(lim_{x \to 2} x + 1) \cdot lim_{x \to 2} 3 x^{2} - 9$

Step 4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$(lim_{x \to 2} x + 1) \cdot (lim_{x \to 2} 3 x^{2} - lim_{x \to 2} 9)$

Step 5

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

$(lim_{x \to 2} x + 1) \cdot (3 lim_{x \to 2} x^{2} - lim_{x \to 2} 9)$

Step 6

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

$(lim_{x \to 2} x + 1) \cdot (3 {(lim_{x \to 2} x)}^{2} - lim_{x \to 2} 9)$

Step 7

Evaluate the limit of $9$ which is constant as $x$ approaches $2$ .

$(lim_{x \to 2} x + 1) \cdot (3 {(lim_{x \to 2} x)}^{2} - 1 \cdot 9)$

Step 8

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 8.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$(2 + 1) \cdot (3 {(lim_{x \to 2} x)}^{2} - 1 \cdot 9)$

Step 8.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$(2 + 1) \cdot (3 \cdot 2^{2} - 1 \cdot 9)$

Step 9

Simplify the answer.

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Step 9.1

Add $2$ and $1$ .

$3 \cdot (3 \cdot 2^{2} - 1 \cdot 9)$

Step 9.2

Simplify each term.

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Step 9.2.1

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

$3 \cdot (3 \cdot 4 - 1 \cdot 9)$

Step 9.2.2

Multiply $3$ by $4$ .

$3 \cdot (12 - 1 \cdot 9)$

Step 9.2.3

Multiply $- 1$ by $9$ .

$3 \cdot (12 - 9)$

Step 9.3

Subtract $9$ from $12$ .

$3 \cdot 3$

Step 9.4

Multiply $3$ by $3$ .

$9$