What is the limit as x approaches 0 of (sqrt(1+7x)-sqrt(1-5x))/x ?

The limit as x approaches 0 of (sqrt(1+7x)-sqrt(1-5x))/x is 6

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limit as x approaches 0 of (sqrt(1+7x)-sqrt(1-5x))/x

x \to 0 lim (\frac{1 + 7 x - 1 - 5 x}{x})

6

Solution steps

x \to 0 lim (\frac{1 + 7 x - 1 - 5 x}{x})

Rationalize numerator

\frac{1 + 7 x - 1 - 5 x}{x} : \frac{12}{7 x + 1 + - 5 x + 1}

= x \to 0 lim (\frac{12}{7 x + 1 + - 5 x + 1})

Plug in the value

x = 0

= \frac{12}{7 \cdot 0 + 1 + - 5 \cdot 0 + 1}

Simplify

\frac{12}{7 \cdot 0 + 1 + - 5 \cdot 0 + 1} : 6

= 6

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What is the limit as x approaches 0 of (sqrt(1+7x)-sqrt(1-5x))/x ?
The limit as x approaches 0 of (sqrt(1+7x)-sqrt(1-5x))/x is 6