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Does the sequence $f_n(x)=e^x(1+x/n)$ converge uniformly on R? What would be the function it converges to if it does?

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it converges obviously simply toward $e^x$ (fix x and make n tends to $+\infty$) the difference is $e^x\frac{x}{n}$ that is obviously not bounded for all x. So you have no uniform convergence.

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