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

The limit as x approaches 0 of (sqrt(4+x)-sqrt(4-x))/(4x) is 1/8

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

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

\frac{1}{8}

Decimal

0.125

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)

= \frac{1}{4} \cdot x \to 0 lim (\frac{4 + x - 4 - x}{x})

Rationalize numerator

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

= \frac{1}{4} \cdot x \to 0 lim (\frac{2}{x + 4 + - x + 4})

Plug in the value

x = 0

= \frac{1}{4} \cdot \frac{2}{0 + 4 + - 0 + 4}

Simplify

\frac{1}{4} \cdot \frac{2}{0 + 4 + - 0 + 4} : \frac{1}{8}

= \frac{1}{8}

What is the limit as x approaches 0 of (sqrt(4+x)-sqrt(4-x))/(4x) ?
The limit as x approaches 0 of (sqrt(4+x)-sqrt(4-x))/(4x) is 1/8