What is the limit as x approaches infinity of (-7x^2-1)/(x^2+sqrt(x)4+7x^2+1) ?

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limit as x approaches infinity of (-7x^2-1)/(x^2+sqrt(x)4+7x^2+1)

x \to \infty lim (\frac{- 7 x ^{2} - 1}{x ^{2} + x 4 + 7 x ^{2} + 1})

- \frac{7}{8}

Decimal

- 0.875

Solution steps

x \to \infty lim (\frac{- 7 x ^{2} - 1}{x ^{2} + x \cdot 4 + 7 x ^{2} + 1})

Divide by highest denominator power:

\frac{- 7 - \frac{1}{x ^{2}}}{1 + \frac{4}{x ^{\frac{3}{2}}} + 7 + \frac{1}{x ^{2}}}

= x \to \infty lim (\frac{- 7 - \frac{1}{x ^{2}}}{1 + \frac{4}{x ^{\frac{3}{2}}} + 7 + \frac{1}{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} ( - 7 - \frac{1}{x ^{2}} )}{lim _{x \to \infty} ( 1 + \frac{4}{x ^{\frac{3}{2}}} + 7 + \frac{1}{x ^{2}} )}

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

x \to \infty lim (1 + \frac{4}{x ^{\frac{3}{2}}} + 7 + \frac{1}{x ^{2}}) = 8

= \frac{- 7}{8}

Simplify

= - \frac{7}{8}

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x \to 0 + lim (sin ((\frac{1}{x})^{x}))

x \to 9 lim (x^{2} - 81)

x \to π lim (\frac{x}{1 + cos ( x )})

What is the limit as x approaches infinity of (-7x^2-1)/(x^2+sqrt(x)4+7x^2+1) ?
The limit as x approaches infinity of (-7x^2-1)/(x^2+sqrt(x)4+7x^2+1) is -7/8