is equal to

Engineering

Mathematics

Introduction to Determinants

Question

$lim_{x \to 0} \frac{e - (1 + 2 x) \frac{1}{2 x}}{x}$ is equal to

e – e²

$- \frac{2}{e}$

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lim_{x \to 0} \frac{e - e^{\frac{\ln (1 + 2 x)}{2 x}}}{x} = lim_{x \to 0} \frac{(- e) \{e^{\frac{\ln (1 + 2 x)}{2 x} - 1} - 1\}}{x}

= lim_{x \to 0} (- e) \cdot \frac{\{\frac{\ln (1 + 2 x)}{2 x} - 1\}}{x} = - e \cdot lim_{x \to 0} \frac{\ln (1 + 2 x) - 2 x}{2 x^{2}}

= - e \cdot lim_{x \to 0} \frac{2 x - \frac{4 x^{2}}{2} + \frac{8 x^{3}}{3} - \dots . . 2 x}{2 x^{2}}

= (- e) \cdot (\frac{- \frac{4}{2}}{2}) = e

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