The function $f(x)=e^{-1/x^2}\sin(1/x)$ is $C^\infty$ on $\mathbb{R}$ (by defining $f(0)=0$). But it oscillates above and below the $x$ axis infinitely many times as $x\to 0$.
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Analysis
The function $f(x)=e^{-1/x^2}\sin(1/x)$ is $C^\infty$ on $\mathbb{R}$ (by defining $f(0)=0$). But it oscillates above and below the $x$ axis infinitely many times as $x\to 0$.