What is the limit as x approaches 3 of (sqrt(12-x)-3)/(1+x-sqrt(x^2+7)) ?

The limit as x approaches 3 of (sqrt(12-x)-3)/(1+x-sqrt(x^2+7)) is -2/3

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limit as x approaches 3 of (sqrt(12-x)-3)/(1+x-sqrt(x^2+7))

x \to 3 lim (\frac{12 - x - 3}{1 + x - x ^{2} + 7})

- \frac{2}{3}

Decimal Notation

- 0.66666 \dots

Solution steps

x \to 3 lim (\frac{12 - x - 3}{1 + x - x ^{2} + 7})

Apply L'Hopital's Rule

: x \to 3 lim (\frac{- \frac{1}{2 - x + 12}}{- \frac{x}{x ^{2} + 7} + 1})

= x \to 3 lim (\frac{- \frac{1}{2 - x + 12}}{- \frac{x}{x ^{2} + 7} + 1})

Plug in the value

x = 3

= \frac{- \frac{1}{2 - 3 + 12}}{- \frac{3}{3 ^{2} + 7} + 1}

Simplify

\frac{- \frac{1}{2 - 3 + 12}}{- \frac{3}{3 ^{2} + 7} + 1} : - \frac{2}{3}

= - \frac{2}{3}

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What is the limit as x approaches 3 of (sqrt(12-x)-3)/(1+x-sqrt(x^2+7)) ?
The limit as x approaches 3 of (sqrt(12-x)-3)/(1+x-sqrt(x^2+7)) is -2/3