Is function $ f(x)= e^{-(x+x^{2})} $ uniformly continuous on $R$

Question

Is function $ f(x)= e^{-( x+ x^{2}) } $ uniformly continuous on $ R $?

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As we know for any function $f$ which is continuous on $R$ and $ \lim_{ x\to \infty} f(x) $ , $ \lim_{x \to - \infty } f(x) $ exist finitely, then $ f$ is uniformly continuous.

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I am confused about the limit of $f$ at $ - \infty $.

Please help.

Answer 1

$g(x)=-(x^2+x)=-(x+\frac{1}{2})^2+\frac{1}{4}$ $\Rightarrow$ $\lim_{x \to -\infty}g(x)=-\infty$

$\lim_{x \to -\infty}g(x)=-\infty$ $\Rightarrow$ $\lim_{x \to -\infty}e^{g(x)}=0$