What is the limit as x approaches-infinity of ((15x^2+7x^{1/3}))/((sqrt(9x^4+7))) ?

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limit as x approaches-infinity of ((15x^2+7x^{1/3}))/((sqrt(9x^4+7)))

x \to - \infty lim \frac{( 15 x ^{2} + 7 x ^{\frac{1}{3}} )}{( 9 x ^{4} + 7 )}

5

Solution steps

x \to - \infty lim (\frac{15 x ^{2} + 7 x ^{\frac{1}{3}}}{9 x ^{4} + 7})

Apply L'Hopital's Rule

: x \to - \infty lim \frac{\frac{7 + 90 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{7 + 90 x ^{\frac{5}{3}}}{3 x ^{\frac{2}{3}}}}{\frac{18 x ^{3}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}

Simplify

\frac{\frac{7 + 90 x ^{\frac{5}{3}}}{3 x ^{\frac{2}{3}}}}{\frac{18 x ^{3}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}} : \frac{( 7 + 90 x ^{\frac{5}{3}} ) ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}{54 x ^{\frac{11}{3}}}

= x \to - \infty lim \frac{( 7 + 90 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{126 x ^{3} + 2970 x ^{\frac{14}{3}} + 1050 x ^{\frac{2}{3}}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}{198 x ^{\frac{8}{3}}}

= x \to - \infty lim \frac{\frac{126 x ^{3} + 2970 x ^{\frac{14}{3}} + 1050 x ^{\frac{2}{3}}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}{198 x ^{\frac{8}{3}}}

Simplify

\frac{\frac{126 x ^{3} + 2970 x ^{\frac{14}{3}} + 1050 x ^{\frac{2}{3}}}{( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}}{198 x ^{\frac{8}{3}}} : \frac{21 x ^{3} + 495 x ^{\frac{14}{3}} + 175 x ^{\frac{2}{3}}}{33 x ^{\frac{8}{3}} ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}}

= x \to - \infty lim (\frac{21 x ^{3} + 495 x ^{\frac{14}{3}} + 175 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{21 x ^{3} + 495 x ^{\frac{14}{3}} + 175 x ^{\frac{2}{3}}}{x ^{\frac{8}{3}} ( 9 x ^{4} + 7 ) ^{\frac{1}{2}}})

Divide by

x^{\frac{14}{3}} : \frac{\frac{21}{x ^{\frac{5}{3}}} + 495 + \frac{175}{x ^{4}}}{( 9 + \frac{7}{x ^{4}} ) ^{\frac{1}{2}}}

= \frac{1}{33} \cdot x \to - \infty lim \frac{\frac{21}{x ^{\frac{5}{3}}} + 495 + \frac{175}{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{21}{x ^{\frac{5}{3}}} + 495 + \frac{175}{x ^{4}} )}{lim _{x \to - \infty} ( ( 9 + \frac{7}{x ^{4}} ) ^{\frac{1}{2}} )}

x \to - \infty lim (\frac{21}{x ^{\frac{5}{3}}} + 495 + \frac{175}{x ^{4}}) = 495

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

= \frac{1}{33} \cdot \frac{495}{3}

Simplify

\frac{1}{33} \cdot \frac{495}{3} : 5

= 5

x \to - 1 - lim (2 x + 1)

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

x \to - \infty lim (x - x^{3})

x \to 5 lim (\frac{x ^{2} - 3 x - 10}{x ^{2} - 4 x - 5})

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

What is the limit as x approaches-infinity of ((15x^2+7x^{1/3}))/((sqrt(9x^4+7))) ?
The limit as x approaches-infinity of ((15x^2+7x^{1/3}))/((sqrt(9x^4+7))) is 5