What is the limit as x approaches-infinity of ((-x+sqrt(x^2-8)))/(6x-7) ?

The limit as x approaches-infinity of ((-x+sqrt(x^2-8)))/(6x-7) is -1/3

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limit as x approaches-infinity of ((-x+sqrt(x^2-8)))/(6x-7)

x \to - \infty lim (\frac{( - x + x ^{2} - 8 )}{6 x - 7})

- \frac{1}{3}

Decimal Notation

- 0.33333 \dots

Solution steps

x \to - \infty lim (\frac{- x + x ^{2} - 8}{6 x - 7})

Divide by highest denominator power:

\frac{- 1 + \frac{x ^{2} - 8}{x}}{6 - \frac{7}{x}}

= x \to - \infty lim (\frac{- 1 + \frac{x ^{2} - 8}{x}}{6 - \frac{7}{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{x ^{2} - 8}{x} )}{lim _{x \to - \infty} ( 6 - \frac{7}{x} )}

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

x \to - \infty lim (6 - \frac{7}{x}) = 6

= \frac{- 2}{6}

Simplify

\frac{- 2}{6} : - \frac{1}{3}

= - \frac{1}{3}

x + 2 sin (x)

x \to 0 lim (3 x)

e^{- 2 (2 t - 4)} (\frac{1}{2} (2 t - 4) + 2)

\int - e^{y} sin (y) cos (y) d y

\int (\frac{1}{z ^{3}} - \frac{3}{z ^{2}}) d z

What is the limit as x approaches-infinity of ((-x+sqrt(x^2-8)))/(6x-7) ?
The limit as x approaches-infinity of ((-x+sqrt(x^2-8)))/(6x-7) is -1/3