Calculus Examples

$\frac{lim_{x \to 3} (\sqrt{x + 6}) - x}{x^{3} - 3 x^{2}}$

Step 1

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

$\frac{lim_{x \to 3} \sqrt{x + 6} - lim_{x \to 3} x}{x^{3} - 3 x^{2}}$

Step 2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 3} x + 6} - lim_{x \to 3} x}{x^{3} - 3 x^{2}}$

Step 3

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

$\frac{\sqrt{lim_{x \to 3} x + lim_{x \to 3} 6} - lim_{x \to 3} x}{x^{3} - 3 x^{2}}$

Step 4

Evaluate the limit of $6$ which is constant as $x$ approaches $3$ .

$\frac{\sqrt{lim_{x \to 3} x + 6} - lim_{x \to 3} x}{x^{3} - 3 x^{2}}$

Step 5

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

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Step 5.1

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

$\frac{\sqrt{3 + 6} - lim_{x \to 3} x}{x^{3} - 3 x^{2}}$

Step 5.2

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

$\frac{\sqrt{3 + 6} - 3}{x^{3} - 3 x^{2}}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Add $3$ and $6$ .

$\frac{\sqrt{9} - 3}{x^{3} - 3 x^{2}}$

Step 6.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2}} - 3}{x^{3} - 3 x^{2}}$

Step 6.1.3

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

$\frac{3 - 3}{x^{3} - 3 x^{2}}$

Step 6.1.4

Subtract $3$ from $3$ .

$\frac{0}{x^{3} - 3 x^{2}}$

Step 6.2

Factor $x^{2}$ out of $x^{3} - 3 x^{2}$ .

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Step 6.2.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{0}{x^{2} x - 3 x^{2}}$

Step 6.2.2

Factor $x^{2}$ out of $- 3 x^{2}$ .

$\frac{0}{x^{2} x + x^{2} \cdot - 3}$

Step 6.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 3$ .

$\frac{0}{x^{2} (x - 3)}$

Step 6.3

Divide $0$ by $x^{2} (x - 3)$ .

$0$