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

The limit as x approaches 1 of (sqrt(x+2)-sqrt(3))/(x^2-1) is 1/(4sqrt(3))

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

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

\frac{1}{4 3}

Decimal

0.14433 \dots

Solution steps

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

Multiply by the conjugate of

x + 2 - 3 : \frac{x - 1}{x + 2 + 3}

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

Simplify

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

= x \to 1 lim (\frac{1}{( x + 2 + 3 ) ( x + 1 )})

Plug in the value

x = 1

= \frac{1}{( 1 + 2 + 3 ) ( 1 + 1 )}

Simplify

= \frac{1}{4 3}

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

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

x \to \infty lim (- 1^{5 x})

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

x \to 6 lim (2 x^{2} + 4)

What is the limit as x approaches 1 of (sqrt(x+2)-sqrt(3))/(x^2-1) ?
The limit as x approaches 1 of (sqrt(x+2)-sqrt(3))/(x^2-1) is 1/(4sqrt(3))