Calculus Examples

$\frac{lim_{x \to 2^{-}} \sqrt{2 - x}}{x^{2} + 5 x}$

Step 1

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 2^{-}} 2 - x}}{x^{2} + 5 x}$

Step 2

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

$\frac{\sqrt{lim_{x \to 2^{-}} 2 - lim_{x \to 2^{-}} x}}{x^{2} + 5 x}$

Step 3

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

$\frac{\sqrt{2 - lim_{x \to 2^{-}} x}}{x^{2} + 5 x}$

Step 4

Simplify terms.

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Step 4.1

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

$\frac{\sqrt{2 - 2}}{x^{2} + 5 x}$

Step 4.2

Simplify the answer.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Subtract $2$ from $2$ .

$\frac{\sqrt{0}}{x^{2} + 5 x}$

Step 4.2.1.2

Rewrite $0$ as $0^{2}$ .

$\frac{\sqrt{0^{2}}}{x^{2} + 5 x}$

Step 4.2.1.3

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

$\frac{0}{x^{2} + 5 x}$

Step 4.2.2

Factor $x$ out of $x^{2} + 5 x$ .

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Step 4.2.2.1

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

$\frac{0}{x \cdot x + 5 x}$

Step 4.2.2.2

Factor $x$ out of $5 x$ .

$\frac{0}{x \cdot x + x \cdot 5}$

Step 4.2.2.3

Factor $x$ out of $x \cdot x + x \cdot 5$ .

$\frac{0}{x (x + 5)}$

Step 4.2.3

Divide $0$ by $x (x + 5)$ .

$0$