limit as x approaches infinity+of (log_{4}(x))/(log_{7)(x)}

解

x \to \infty + lim (\frac{lo g _{4} ( x )}{lo g _{7} ( x )})

\frac{ln ( 7 )}{2 ln ( 2 )}

十進法表記

1.40367 \dots

解答ステップ

x \to \infty lim (\frac{lo g _{4} ( x )}{lo g _{7} ( x )})

ロピタルの定理を適用する

= x \to \infty lim (\frac{\frac{1}{2 l n ( 2 ) x}}{\frac{1}{x l n ( 7 )}})

\frac{\frac{1}{2 l n ( 2 ) x}}{\frac{1}{x l n ( 7 )}} = \frac{ln ( 7 )}{2 ln ( 2 )}

= x \to \infty lim (\frac{ln ( 7 )}{2 ln ( 2 )})

x \to a lim c = c

= \frac{ln ( 7 )}{2 ln ( 2 )}