What is the limit as x approaches 0 of (4x)/(sqrt(4+x)-\sqrt{4-x)} ?

The limit as x approaches 0 of (4x)/(sqrt(4+x)-\sqrt{4-x)} is 8

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limit as x approaches 0 of (4x)/(sqrt(4+x)-\sqrt{4-x)}

x \to 0 lim (\frac{4 x}{4 + x - 4 - x})

8

Solution steps

x \to 0 lim (\frac{4 x}{4 + x - 4 - x})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 4 \cdot x \to 0 lim (\frac{x}{4 + x - 4 - x})

Rationalize denominator

\frac{x}{4 + x - 4 - x} : \frac{x + 4 + - x + 4}{2}

= 4 \cdot x \to 0 lim (\frac{x + 4 + - x + 4}{2})

Plug in the value

x = 0

= 4 \cdot \frac{0 + 4 + - 0 + 4}{2}

Simplify

4 \cdot \frac{0 + 4 + - 0 + 4}{2} : 8

= 8

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What is the limit as x approaches 0 of (4x)/(sqrt(4+x)-\sqrt{4-x)} ?
The limit as x approaches 0 of (4x)/(sqrt(4+x)-\sqrt{4-x)} is 8