What is the limit as x approaches infinity of (sqrt(3x^8-17x^3+5))/(8x^4+12x^3-x) ?

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limit as x approaches infinity of (sqrt(3x^8-17x^3+5))/(8x^4+12x^3-x)

x \to \infty lim (\frac{3 x ^{8} - 17 x ^{3} + 5}{8 x ^{4} + 12 x ^{3} - x})

\frac{3}{8}

Decimal

0.21650 \dots

Solution steps

x \to \infty lim (\frac{3 x ^{8} - 17 x ^{3} + 5}{8 x ^{4} + 12 x ^{3} - x})

Divide by highest denominator power:

\frac{3 - \frac{17}{x ^{5}} + \frac{5}{x ^{8}}}{8 + \frac{12}{x} - \frac{1}{x ^{3}}}

= x \to \infty lim \frac{3 - \frac{17}{x ^{5}} + \frac{5}{x ^{8}}}{8 + \frac{12}{x} - \frac{1}{x ^{3}}}

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} ( 3 - \frac{17}{x ^{5}} + \frac{5}{x ^{8}} )}{lim _{x \to \infty} ( 8 + \frac{12}{x} - \frac{1}{x ^{3}} )}

x \to \infty lim (3 - \frac{17}{x ^{5}} + \frac{5}{x ^{8}}) = 3

x \to \infty lim (8 + \frac{12}{x} - \frac{1}{x ^{3}}) = 8

= \frac{3}{8}

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

x \to 0 lim (\frac{∣ x ∣}{3 x})

x \to 4 lim (\frac{\frac{2}{x + 1} - \frac{4}{x + 6}}{x + 4})

x \to 0 lim (\frac{2}{- sin ( x )})

x \to - \frac{5}{2} lim (\frac{( 8 x ^{3} + 125 )}{6 x ^{2} + 23 x + 20})

What is the limit as x approaches infinity of (sqrt(3x^8-17x^3+5))/(8x^4+12x^3-x) ?
The limit as x approaches infinity of (sqrt(3x^8-17x^3+5))/(8x^4+12x^3-x) is (sqrt(3))/8