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

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

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limit as x approaches infinity of x^{((ln(3))/(1+ln(x)))}

x \to \infty lim (x^{(\frac{l n ( 3 )}{1 + l n ( x )})})

3

Solution steps

x \to \infty lim (x^{\frac{l n ( 3 )}{1 + l n ( x )}})

Apply exponent rules

= x \to \infty lim (e^{\frac{l n ( 3 )}{1 + l n ( x )} l n (x)})

Apply the Limit Chain Rule:

3

= 3

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

x \to 1 - lim (x - 1)

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

h \to 0 lim (\frac{h ∣ h ∣}{h})

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

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