What is the limit as x approaches 1 of (sqrt(10-x)-sqrt(8+x))/(x^3-1) ?

The limit as x approaches 1 of (sqrt(10-x)-sqrt(8+x))/(x^3-1) is -1/9

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limit as x approaches 1 of (sqrt(10-x)-sqrt(8+x))/(x^3-1)

x \to 1 lim (\frac{10 - x - 8 + x}{x ^{3} - 1})

- \frac{1}{9}

Decimal

- 0.11111 \dots

Solution steps

x \to 1 lim (\frac{10 - x - 8 + x}{x ^{3} - 1})

Rationalize numerator

\frac{10 - x - 8 + x}{x ^{3} - 1} : - \frac{2}{( x ^{2} + x + 1 ) ( - x + 10 + x + 8 )}

= x \to 1 lim (- \frac{2}{( x ^{2} + x + 1 ) ( - x + 10 + x + 8 )})

Plug in the value

x = 1

= - \frac{2}{( 1 ^{2} + 1 + 1 ) ( - 1 + 10 + 1 + 8 )}

Simplify

- \frac{2}{( 1 ^{2} + 1 + 1 ) ( - 1 + 10 + 1 + 8 )} : - \frac{1}{9}

= - \frac{1}{9}

What is the limit as x approaches 1 of (sqrt(10-x)-sqrt(8+x))/(x^3-1) ?
The limit as x approaches 1 of (sqrt(10-x)-sqrt(8+x))/(x^3-1) is -1/9