Find the value of the given limit: [duplicate]

real-analysis limits functions integration uniform-convergence

$$\lim_{n\rightarrow\infty}\int_0^1\frac{ne^x}{1+n^2x^2} dx$$ I deduced that the sequence of function is not uniformly convergent to it's functional limit i.e. $f(x)=0$ but couldn't proceed further to calculate the final value. Can someone please guide me on how to proceed?


Hint :

  1. Show that$$\int_0^1\frac{ne^x}{1+n^2x^2} dx = \int_0^{+\infty}\frac{e^{x/n}}{1+x^2} \chi_{[0,n]} dx $$

  2. Show that for every $x \geq 0$,

$$\left| \frac{e^{x/n}}{1+x^2} \chi_{[0,n]} \right| \leq \frac{e}{1+x^2}$$

  1. Use the dominated convergence theorem to conclude that $$\lim_{n \rightarrow +\infty} \int_0^1\frac{ne^x}{1+n^2x^2} dx = \frac{\pi}{2}$$

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