What is the limit as x approaches 1 of (ln(x))/(9^{ln(x))-x} ?

The limit as x approaches 1 of (ln(x))/(9^{ln(x))-x} is 1/(2ln(3)-1)

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limit as x approaches 1 of (ln(x))/(9^{ln(x))-x}

x \to 1 lim (\frac{ln ( x )}{9 ^{l n (x)} - x})

\frac{1}{2 ln ( 3 ) - 1}

Decimal

0.83526 \dots

Solution steps

x \to 1 lim (\frac{ln ( x )}{9 ^{l n (x)} - x})

Apply L'Hopital's Rule

: x \to 1 lim (\frac{\frac{1}{x}}{2 ln ( 3 ) x ^{2 l n (3) - 1} - 1})

= x \to 1 lim (\frac{\frac{1}{x}}{2 ln ( 3 ) x ^{2 l n (3) - 1} - 1})

Plug in the value

x = 1

= \frac{\frac{1}{1}}{2 ln ( 3 ) \cdot 1 ^{2 l n (3) - 1} - 1}

Simplify

= \frac{1}{2 ln ( 3 ) - 1}

x \to - \infty lim (x lo g_{e} (1 - \frac{1}{x}))

x \to 0 lim (2 - cot (x))

x \to \infty lim (1 - 3 x)

x \to - π lim ((x - π) csc (x))

x \to 5 lim (3 x - 12)

What is the limit as x approaches 1 of (ln(x))/(9^{ln(x))-x} ?
The limit as x approaches 1 of (ln(x))/(9^{ln(x))-x} is 1/(2ln(3)-1)