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

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

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

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

- \frac{1}{2}

Decimal

- 0.70710 \dots

Solution steps

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

Divide by highest denominator power:

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

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

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

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

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

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

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

= - \frac{2}{2}

Simplify

- \frac{2}{2} : - \frac{1}{2}

= - \frac{1}{2}

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