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&= 1 + \left( 1 - \fracx2 +o(x) - 1 \right) + o \left( 1 - \fracx2 + o(x) - 1 \right) \\<9pt> &= 1 - \fracx2 + o(x). \endalign*" title="Rendered by QuickLaTeX.com"/>

Therefore, we have

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&= e \cdot \lim_x \to 0 \frace^\frac1x \log(1+x) - 1 - 1x \\<9pt> &= e \cdot \lim_x \to 0 \frac1x \left( 1 - \fracx2 + o(x) - 1 \right) \\<9pt> &= e \cdot \lim_x \to 0 \left( -\frac12 + \fraco(x)x \right) \\<9pt> &= -\frace2. \endalign*" title="Rendered by QuickLaTeX.com"/>




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If you are having trouble with math proofs a great book to learn from is How lớn Prove It by Daniel Velleman:


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A really awesome book that I highly recommend on how to lớn study math and be a math major is Laura Alcock"s, How to Study as a Mathematics Major: