Hi, I was working on this problem on Khan Academy and am wondering how they got the answer they did.

. . .[tex]\displaystyle \lim_{x \rightarrow 1}\, \dfrac{x\, e^x\, -\, e}{x^2\, -\, 1}[/tex]

. . . . . .[tex]\displaystyle =\, \lim_{x \rightarrow 1}\, \dfrac{\dfrac{d}{dx}\left[x\, e^x\, -\, e\right]}{\dfrac{d}{dx}\left[x^2\, -\, 1\right]}\qquad \mbox{ L'Hopital's rule}[/tex]

. . . . . .[tex]\displaystyle =\,\lim_{x \rightarrow 1}\, \dfrac{e^x\, +\, x\, e^x}{2x}[/tex]

. . . . . .[tex]\displaystyle =\, \dfrac{e^{(1)}\, +\, (1)\, e^{(1)}}{2(1)}\qquad \mbox{ Substitution}[/tex]

. . . . . .[tex]\displaystyle =\, e[/tex]

My problem with their solution is how the negative e suddenly gained an x

exponent (I know e stays the same after a derivative so is it just symbolic?) and also how it became positive.

Any help would be greatly appreciated.

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