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初等函数不等式

\[ \begin{array}{*{20}c} {\sin x < x < \tan x} & {\left( {0 < x < \frac{\pi }{2}} \right)} \\ {\cos x < \frac{{\sin x}}{x} < 1} & {\left( {0 < x < \pi } \right)} \\ {\frac{{\sin x}}{x} > \frac{2}{\pi }} & {\left( { - \frac{\pi }{2} < x < \frac{\pi }{2}} \right)} \\ {\cos x > 1 - \frac{1}{2}x^2 } & {\left( { - \infty < x < \infty ,x \ne 0} \right)} \\ {\sin x > x - \frac{1}{6}x^3 } & {\left( {x > 0} \right)} \\ {\tan x > x + \frac{1}{3}x^3 } & {\left( {0 < x < \frac{\pi }{2}} \right)} \\ {\pi < \frac{{\sin \pi x}}{{x\left( {1 - x} \right)}} < 4} & {\left( {0 < x < 1,x \ne \frac{1}{2}} \right)} \\ \end{array} \] \[ \begin{array}{*{20}c} {e^x \, > 1 + x} & {\left( {x \ne 0} \right)} \\ {e^x < \frac{1}{{1 - x}}} & {\left( {x < 1,x \ne 0} \right)} \\ {e^x > 1 + x + \frac{{x^2 }}{{2!}} + \cdots + \frac{{x^n }}{{n!}}} & {\left( {n为正整数,x > 0} \right)} \\ {e^{\frac{x}{{1 - x}}} > \frac{1}{{1 - x}}} & {\left( {x \ne 0} \right)} \\ {\frac{x}{{1 - x}} < 1 - e^{ - x} } & {\left( {x > - 1,x \ne 0} \right)} \\ {e^{\frac{x}{{1 + x}}} < 1 + x} & {\left( {x > - 1,x \ne 0} \right)} \\ {\frac{x}{{1 + x}} < \ln \left( {1 + x} \right) < x} & {\left( {x > - 1,x \ne 0} \right)} \\ \end{array} \] 特别取 \[ x = \frac{1}{n}\left( n为正整数 \right) \] 有 \[ \frac{1}{{n + 1}} < \ln \left( {1 + \frac{1}{n}} \right) < \frac{1}{n} \]

\[ \begin{array}{*{20}c} {\ln x \le x - 1} & {\left( {x > 0} \right)} \\ {x < - \ln \left( {1 - x} \right) < \frac{x}{{1 - x}}} & {\left( {x < 1,x \ne 0} \right)} \\ {\ln x \le n\left( {x^{\frac{1}{n}} - 1} \right)} & {\left( {n > 0,x > 0} \right)} \\ {\ln \sec x < \frac{1}{2}\sin x \cdot \tan x} & {\left( {0 < x < \frac{1}{2}} \right)} \\ {\left( {1 + x} \right)^\alpha > 1 + x^\alpha } & {\left( {\alpha > 1,x > 0} \right)} \\ \end{array} \]