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

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

x \to \infty lim (\frac{3 x ^{6} + 3 x ^{2} + 2}{7 x ^{6} + x - 1})

\frac{3}{7}

Decimal

0.42857 \dots

Solution steps

x \to \infty lim (\frac{3 x ^{6} + 3 x ^{2} + 2}{7 x ^{6} + x - 1})

Divide by highest denominator power:

\frac{3 + \frac{3}{x ^{4}} + \frac{2}{x ^{6}}}{7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}}}

= x \to \infty lim (\frac{3 + \frac{3}{x ^{4}} + \frac{2}{x ^{6}}}{7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}}})

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{3}{x ^{4}} + \frac{2}{x ^{6}} )}{lim _{x \to \infty} ( 7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}} )}

x \to \infty lim (3 + \frac{3}{x ^{4}} + \frac{2}{x ^{6}}) = 3

x \to \infty lim (7 + \frac{1}{x ^{5}} - \frac{1}{x ^{6}}) = 7

= \frac{3}{7}

x \to 6 lim (\frac{( sin ( x ) )}{x - 3})

x \to 4 lim (\frac{3 - x}{x ^{2} - 9})

x \to - 2 lim (\frac{( 8 x ^{3} - 16 x ^{2} - 64 x )}{x ^{2} + 2 x})

x \to \infty lim (10 x (ln (x + 8) - ln (x)))

x \to 0 lim (\frac{5}{x} - \frac{5}{e ^{t} - 1})

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