Question

$\lim_{x \to \infty} ln(1+e^x)-x=?$

Answer 1

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Answer 2

[tex] \lim\limits_{x \to \infty} \Big[\ln (1+ e^x) -x\Big] = \\ \\ = \lim\limits_{x \to \infty} \Big[\ln(1+e^x) - \ln e^x\Big] = \\ \\ = \lim\limits_{x \to \infty}\ln \Big(\dfrac{1 + e^x}{e^x} \Big)= \\ \\ =\ln\Big[\lim\limits_{x \to \infty} \Big(\dfrac{ 1 + e ^x}{e ^ x} \Big)\Big]= \\ \\ = \ln\Big[ \lim\limits_{x \to \infty} \Big( \dfrac{ 1}{e ^ x} + 1 \Big) \Big]= \\ \\ = \ln 1 = 0 [/tex]