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

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

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

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

\frac{2}{3}

Decimal Notation

0.66666 \dots

Solution steps

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

Rationalize denominator

\frac{x + 2}{2 x + x ^{2} + 12} : \frac{2 x - x ^{2} + 12}{3 ( x - 2 )}

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

Plug in the value

x = - 2

= \frac{2 ( - 2 ) - ( - 2 ) ^{2} + 12}{3 ( - 2 - 2 )}

Simplify

\frac{2 ( - 2 ) - ( - 2 ) ^{2} + 12}{3 ( - 2 - 2 )} : \frac{2}{3}

= \frac{2}{3}

x \to 3 lim (\frac{( 2 x ^{3} - 5 x ^{2} - 2 x - 3 )}{4 x ^{3} - 13 x ^{2} + 4 x - 3})

x \to 0 lim (x \cdot cos (x))

\int \frac{- 4 x + 4}{5 ( 4 x ^{2} - 4 x + 5 )} d x

f (θ) = 4 sin (θ)

y = \frac{14}{x}, y = - 64 x + 72

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