Precalculus Examples

$lim_{x \to 3} (x^{4} - 3) (x^{2} + 5 x + 3)$

Step 1

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

$lim_{x \to 3} x^{4} - 3 \cdot lim_{x \to 3} x^{2} + 5 x + 3$

Step 2

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

$(lim_{x \to 3} x^{4} - lim_{x \to 3} 3) \cdot lim_{x \to 3} x^{2} + 5 x + 3$

Step 3

Move the exponent $4$ from $x^{4}$ outside the limit using the Limits Power Rule.

$({(lim_{x \to 3} x)}^{4} - lim_{x \to 3} 3) \cdot lim_{x \to 3} x^{2} + 5 x + 3$

Step 4

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

$({(lim_{x \to 3} x)}^{4} - 1 \cdot 3) \cdot lim_{x \to 3} x^{2} + 5 x + 3$

Step 5

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

$({(lim_{x \to 3} x)}^{4} - 1 \cdot 3) \cdot (lim_{x \to 3} x^{2} + lim_{x \to 3} 5 x + lim_{x \to 3} 3)$

Step 6

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

$({(lim_{x \to 3} x)}^{4} - 1 \cdot 3) \cdot ({(lim_{x \to 3} x)}^{2} + lim_{x \to 3} 5 x + lim_{x \to 3} 3)$

Step 7

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

$({(lim_{x \to 3} x)}^{4} - 1 \cdot 3) \cdot ({(lim_{x \to 3} x)}^{2} + 5 lim_{x \to 3} x + lim_{x \to 3} 3)$

Step 8

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

$({(lim_{x \to 3} x)}^{4} - 1 \cdot 3) \cdot ({(lim_{x \to 3} x)}^{2} + 5 lim_{x \to 3} x + 3)$

Step 9

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

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Step 9.1

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

$(3^{4} - 1 \cdot 3) \cdot ({(lim_{x \to 3} x)}^{2} + 5 lim_{x \to 3} x + 3)$

Step 9.2

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

$(3^{4} - 1 \cdot 3) \cdot (3^{2} + 5 lim_{x \to 3} x + 3)$

Step 9.3

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

$(3^{4} - 1 \cdot 3) \cdot (3^{2} + 5 \cdot 3 + 3)$

Step 10

Simplify the answer.

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Step 10.1

Simplify each term.

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Step 10.1.1

Raise $3$ to the power of $4$ .

$(81 - 1 \cdot 3) \cdot (3^{2} + 5 \cdot 3 + 3)$

Step 10.1.2

Multiply $- 1$ by $3$ .

$(81 - 3) \cdot (3^{2} + 5 \cdot 3 + 3)$

Step 10.2

Subtract $3$ from $81$ .

$78 \cdot (3^{2} + 5 \cdot 3 + 3)$

Step 10.3

Simplify each term.

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Step 10.3.1

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

$78 \cdot (9 + 5 \cdot 3 + 3)$

Step 10.3.2

Multiply $5$ by $3$ .

$78 \cdot (9 + 15 + 3)$

Step 10.4

Add $9$ and $15$ .

$78 \cdot (24 + 3)$

Step 10.5

Add $24$ and $3$ .

$78 \cdot 27$

Step 10.6

Multiply $78$ by $27$ .

$2106$