What is the limit as x approaches infinity of (ln(1+e^x))/(12x+11) ?

The limit as x approaches infinity of (ln(1+e^x))/(12x+11) is 1/12

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limit as x approaches infinity of (ln(1+e^x))/(12x+11)

x \to \infty lim (\frac{ln ( 1 + e ^{x} )}{12 x + 11})

\frac{1}{12}

Decimal Notation

0.08333 \dots

Solution steps

x \to \infty lim (\frac{ln ( 1 + e ^{x} )}{12 x + 11})

Divide by highest denominator power:

\frac{\frac{l n ( 1 + e ^{x} )}{x}}{12 + \frac{11}{x}}

= x \to \infty lim (\frac{\frac{l n ( 1 + e ^{x} )}{x}}{12 + \frac{11}{x}})

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{lim _{x \to \infty} ( \frac{l n ( 1 + e ^{x} )}{x} )}{lim _{x \to \infty} ( 12 + \frac{11}{x} )}

x \to \infty lim (\frac{ln ( 1 + e ^{x} )}{x}) = 1

x \to \infty lim (12 + \frac{11}{x}) = 12

= \frac{1}{12}

x \to 1 lim (x^{2} + 6 x)

x \to 2 + lim (x^{2} + 3)

x \to 2 lim (3 x^{2} + 7 x)

x \to 2 + lim (x + 1)

x \to 1 lim (\frac{0}{x})

What is the limit as x approaches infinity of (ln(1+e^x))/(12x+11) ?
The limit as x approaches infinity of (ln(1+e^x))/(12x+11) is 1/12