What is the limit as x approaches-infinity of (sqrt(x^2+8x))/(8x+1) ?

The limit as x approaches-infinity of (sqrt(x^2+8x))/(8x+1) is -1/8

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limit as x approaches-infinity of (sqrt(x^2+8x))/(8x+1)

x \to - \infty lim (\frac{x ^{2} + 8 x}{8 x + 1})

- \frac{1}{8}

Decimal

- 0.125

Solution steps

x \to - \infty lim (\frac{x ^{2} + 8 x}{8 x + 1})

Divide by highest denominator power:

\frac{- 1 + \frac{8}{x}}{8 + \frac{1}{x}}

= x \to - \infty lim \frac{- 1 + \frac{8}{x}}{8 + \frac{1}{x}}

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

= - x \to - \infty lim \frac{1 + \frac{8}{x}}{8 + \frac{1}{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} ( 1 + \frac{8}{x} )}{lim _{x \to - \infty} ( 8 + \frac{1}{x} )}

x \to - \infty lim (1 + \frac{8}{x}) = 1

x \to - \infty lim (8 + \frac{1}{x}) = 8

= - \frac{1}{8}

What is the limit as x approaches-infinity of (sqrt(x^2+8x))/(8x+1) ?
The limit as x approaches-infinity of (sqrt(x^2+8x))/(8x+1) is -1/8