Natural logarithm

Question

Can someone please suggest how one proves:

$(1+2x)\ln(1+\frac{1}{x}) -2 >0$ where $x>0$.

I plotted the function in a program and the inequality should be correct.

Answer 1

Here is one approach. Let $f(x)=(1+2x)\ln(1+\frac{1}{x})$. You can show that $\lim\limits_{x\to+\infty}f(x)=2$, for example using l'Hôpital's rule, with a rewrite as $\dfrac{\ln\left(1+\frac{1}{x}\right)}{\left(\dfrac{1}{1+2x}\right)}$. You can show that $f$ is always decreasing by showing that $f'$ is always negative. In case working directly with $f'$ is too cumbersome, it turns out to be easy to show that $f''$ is always positive, and then it would suffice to show that $\lim\limits_{x\to+\infty}f'(x)=0$.