What is the limit as x approaches-infinity of (6x^2+5x^{1/3})/(sqrt(9x^4+7)) ?

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limit as x approaches-infinity of (6x^2+5x^{1/3})/(sqrt(9x^4+7))

x \to - \infty lim (\frac{6 x ^{2} + 5 x ^{\frac{1}{3}}}{9 x ^{4} + 7})

2

Solution steps

x \to - \infty lim (\frac{6 x ^{2} + 5 x ^{\frac{1}{3}}}{9 x ^{4} + 7})

Apply L'Hopital's Rule

: x \to - \infty lim \frac{\frac{5 + 36 x ^{\frac{5}{3}}}{3 x ^{\frac{2}{3}}}}{\frac{18 x ^{3}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}

= x \to - \infty lim \frac{\frac{5 + 36 x ^{\frac{5}{3}}}{3 x ^{\frac{2}{3}}}}{\frac{18 x ^{3}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}

Simplify

\frac{\frac{5 + 36 x ^{\frac{5}{3}}}{3 x ^{\frac{2}{3}}}}{\frac{18 x ^{3}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}} : \frac{( 5 + 36 x ^{\frac{5}{3}} ) ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}{54 x ^{\frac{11}{3}}}

= x \to - \infty lim \frac{( 5 + 36 x ^{\frac{5}{3}} ) ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}{54 x ^{\frac{11}{3}}}

Apply L'Hopital's Rule

: x \to - \infty lim \frac{\frac{90 x ^{3} + 1188 x ^{\frac{14}{3}} + 420 x ^{\frac{2}{3}}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}{198 x ^{\frac{8}{3}}}

= x \to - \infty lim \frac{\frac{90 x ^{3} + 1188 x ^{\frac{14}{3}} + 420 x ^{\frac{2}{3}}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}{198 x ^{\frac{8}{3}}}

Simplify

\frac{\frac{90 x ^{3} + 1188 x ^{\frac{14}{3}} + 420 x ^{\frac{2}{3}}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}{198 x ^{\frac{8}{3}}} : \frac{15 x ^{3} + 198 x ^{\frac{14}{3}} + 70 x ^{\frac{2}{3}}}{33 x ^{\frac{8}{3}} ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}

= x \to - \infty lim (\frac{15 x ^{3} + 198 x ^{\frac{14}{3}} + 70 x ^{\frac{2}{3}}}{33 x ^{\frac{8}{3}} ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}})

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

= \frac{1}{33} \cdot x \to - \infty lim (\frac{15 x ^{3} + 198 x ^{\frac{14}{3}} + 70 x ^{\frac{2}{3}}}{x ^{\frac{8}{3}} ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}})

Divide by

x^{\frac{14}{3}} : \frac{\frac{15}{x ^{\frac{5}{3}}} + 198 + \frac{70}{x ^{4}}}{( 9 + \frac{7}{x ^{4}} ) ^{\frac{1}{2}}}

= \frac{1}{33} \cdot x \to - \infty lim \frac{\frac{15}{x ^{\frac{5}{3}}} + 198 + \frac{70}{x ^{4}}}{( 9 + \frac{7}{x ^{4}} ) ^{\frac{1}{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{1}{33} \cdot \frac{lim _{x \to - \infty} ( \frac{15}{x ^{\frac{5}{3}}} + 198 + \frac{70}{x ^{4}} )}{lim _{x \to - \infty} ( ( 9 + \frac{7}{x ^{4}} ) ^{\frac{1}{2}} )}

x \to - \infty lim (\frac{15}{x ^{\frac{5}{3}}} + 198 + \frac{70}{x ^{4}}) = 198

x \to - \infty lim ((9 + \frac{7}{x ^{4}})^{\frac{1}{2}}) = 3

= \frac{1}{33} \cdot \frac{198}{3}

Simplify

\frac{1}{33} \cdot \frac{198}{3} : 2

= 2

n = 1 \sum \infty n^{n} x^{n}

y^{^{'}} + 8 (tan (8 x)) y = - 5 cos (8 x)

y = x^{2} + 7

\frac{\partial}{\partial x} (\frac{1}{cos ( x )})

What is the limit as x approaches-infinity of (6x^2+5x^{1/3})/(sqrt(9x^4+7)) ?
The limit as x approaches-infinity of (6x^2+5x^{1/3})/(sqrt(9x^4+7)) is 2