What is the limit as x approaches-infinity of (sqrt(3x^2+6))/(5-2x) ?

The limit as x approaches-infinity of (sqrt(3x^2+6))/(5-2x) is (sqrt(3))/2

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limit as x approaches-infinity of (sqrt(3x^2+6))/(5-2x)

x \to - \infty lim (\frac{3 x ^{2} + 6}{5 - 2 x})

\frac{3}{2}

Decimal

0.86602 \dots

Solution steps

x \to - \infty lim (\frac{3 x ^{2} + 6}{5 - 2 x})

Divide by highest denominator power:

\frac{\frac{3 x ^{2} + 6}{x}}{\frac{5}{x} - 2}

= x \to - \infty lim (\frac{\frac{3 x ^{2} + 6}{x}}{\frac{5}{x} - 2})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to - \infty} ( \frac{3 x ^{2} + 6}{x} )}{lim _{x \to - \infty} ( \frac{5}{x} - 2 )}

x \to - \infty lim (\frac{3 x ^{2} + 6}{x}) = - 3

x \to - \infty lim (\frac{5}{x} - 2) = - 2

= \frac{- 3}{- 2}

Simplify

= \frac{3}{2}

What is the limit as x approaches-infinity of (sqrt(3x^2+6))/(5-2x) ?
The limit as x approaches-infinity of (sqrt(3x^2+6))/(5-2x) is (sqrt(3))/2