What is the limit as x approaches infinity of ((x^3)/(x^3+4))^{x^3} ?

The limit as x approaches infinity of ((x^3)/(x^3+4))^{x^3} is 1/(e^4)

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limit as x approaches infinity of ((x^3)/(x^3+4))^{x^3}

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

\frac{1}{e ^{4}}

Decimal Notation

0.01831 \dots

Solution steps

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

Apply exponent rules

= x \to \infty lim (e^{x^{3} l n (\frac{x ^{3}}{x ^{3} + 4})})

Apply the Limit Chain Rule:

\frac{1}{e ^{4}}

= \frac{1}{e ^{4}}

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

x \to 0 lim (\frac{sin ^{2} ( 9 x )}{x})

x \to 0 - lim (x^{2} ln (- x))

x \to 0 + lim (x^{3} e^{\frac{1}{x}})

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

What is the limit as x approaches infinity of ((x^3)/(x^3+4))^{x^3} ?
The limit as x approaches infinity of ((x^3)/(x^3+4))^{x^3} is 1/(e^4)