Calculus Examples

$lim_{x \to 9} \sqrt{4 x^{3} - 1827}$

Step 1

Evaluate the limit.

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Step 1.1

Move the limit under the radical sign.

$\sqrt{lim_{x \to 9} 4 x^{3} - 1827}$

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $9$ .

$\sqrt{lim_{x \to 9} 4 x^{3} - lim_{x \to 9} 1827}$

Step 1.3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\sqrt{4 lim_{x \to 9} x^{3} - lim_{x \to 9} 1827}$

Step 1.4

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

$\sqrt{4 {(lim_{x \to 9} x)}^{3} - lim_{x \to 9} 1827}$

Step 1.5

Evaluate the limit of $1827$ which is constant as $x$ approaches $9$ .

$\sqrt{4 {(lim_{x \to 9} x)}^{3} - 1 \cdot 1827}$

Step 2

Evaluate the limit of $x$ by plugging in $9$ for $x$ .

$\sqrt{4 \cdot 9^{3} - 1 \cdot 1827}$

Step 3

Simplify the answer.

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Step 3.1

Raise $9$ to the power of $3$ .

$\sqrt{4 \cdot 729 - 1 \cdot 1827}$

Step 3.2

Multiply $4$ by $729$ .

$\sqrt{2916 - 1 \cdot 1827}$

Step 3.3

Multiply $- 1$ by $1827$ .

$\sqrt{2916 - 1827}$

Step 3.4

Subtract $1827$ from $2916$ .

$\sqrt{1089}$

Step 3.5

Rewrite $1089$ as $33^{2}$ .

$\sqrt{33^{2}}$

Step 3.6

Pull terms out from under the radical, assuming positive real numbers.

$33$