Calculus Examples

$\frac{lim_{x \to 0} \sqrt{5 x + 4} - 2}{x}$

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sqrt{5 x + 4} - lim_{x \to 0} 2}{x}$

Step 1.2

Move the limit under the radical sign.

$\frac{\sqrt{lim_{x \to 0} 5 x + 4} - lim_{x \to 0} 2}{x}$

Step 1.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{\sqrt{lim_{x \to 0} 5 x + lim_{x \to 0} 4} - lim_{x \to 0} 2}{x}$

Step 1.4

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

$\frac{\sqrt{5 lim_{x \to 0} x + lim_{x \to 0} 4} - lim_{x \to 0} 2}{x}$

Step 1.5

Evaluate the limit of $4$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{5 lim_{x \to 0} x + 4} - lim_{x \to 0} 2}{x}$

Step 1.6

Evaluate the limit of $2$ which is constant as $x$ approaches $0$ .

$\frac{\sqrt{5 lim_{x \to 0} x + 4} - 1 \cdot 2}{x}$

Step 2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sqrt{5 \cdot 0 + 4} - 1 \cdot 2}{x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $5$ by $0$ .

$\frac{\sqrt{0 + 4} - 1 \cdot 2}{x}$

Step 3.1.2

Add $0$ and $4$ .

$\frac{\sqrt{4} - 1 \cdot 2}{x}$

Step 3.1.3

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}} - 1 \cdot 2}{x}$

Step 3.1.4

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

$\frac{2 - 1 \cdot 2}{x}$

Step 3.1.5

Multiply $- 1$ by $2$ .

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

Step 3.1.6

Subtract $2$ from $2$ .

$\frac{0}{x}$

Step 3.2

Divide $0$ by $x$ .

$0$