Precalculus Examples

$lim_{x \to 2} 2 x^{3} + x^{2} + 7$

Step 1

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

$lim_{x \to 2} 2 x^{3} + lim_{x \to 2} x^{2} + lim_{x \to 2} 7$

Step 2

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

$2 lim_{x \to 2} x^{3} + lim_{x \to 2} x^{2} + lim_{x \to 2} 7$

Step 3

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

$2 {(lim_{x \to 2} x)}^{3} + lim_{x \to 2} x^{2} + lim_{x \to 2} 7$

Step 4

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

$2 {(lim_{x \to 2} x)}^{3} + {(lim_{x \to 2} x)}^{2} + lim_{x \to 2} 7$

Step 5

Evaluate the limit of $7$ which is constant as $x$ approaches $2$ .

$2 {(lim_{x \to 2} x)}^{3} + {(lim_{x \to 2} x)}^{2} + 7$

Step 6

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

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Step 6.1

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

$2 \cdot 2^{3} + {(lim_{x \to 2} x)}^{2} + 7$

Step 6.2

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

$2 \cdot 2^{3} + 2^{2} + 7$

Step 7

Simplify the answer.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $2$ by $2^{3}$ by adding the exponents.

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Step 7.1.1.1

Multiply $2$ by $2^{3}$ .

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Step 7.1.1.1.1

Raise $2$ to the power of $1$ .

$2^{1} \cdot 2^{3} + 2^{2} + 7$

Step 7.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + 3} + 2^{2} + 7$

Step 7.1.1.2

Add $1$ and $3$ .

$2^{4} + 2^{2} + 7$

Step 7.1.2

Raise $2$ to the power of $4$ .

$16 + 2^{2} + 7$

Step 7.1.3

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

$16 + 4 + 7$

Step 7.2

Add $16$ and $4$ .

$20 + 7$

Step 7.3

Add $20$ and $7$ .

$27$