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

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

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

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

- 5

Decimal

- 2.23606 \dots

Solution steps

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

Divide by highest denominator power:

\frac{- 5 - \frac{2}{x ^{2}}}{1 + \frac{3}{x}}

= x \to - \infty lim \frac{- 5 - \frac{2}{x ^{2}}}{1 + \frac{3}{x}}

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

= - x \to - \infty lim \frac{5 - \frac{2}{x ^{2}}}{1 + \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} ( 5 - \frac{2}{x ^{2}} )}{lim _{x \to - \infty} ( 1 + \frac{3}{x} )}

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

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

= - \frac{5}{1}

Simplify

= - 5

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