Limits & Continuity: lim_xto0frace-(1+2x)^(1)/(2x)x is equal to | JEE Main 2024 | Yukti Vault

Yukti Vault

JEE Main 2024 (Online) 9th April Evening ShiftQuestion 1

$\lim_{x\to0}\frac{e-(1+2x)^{\frac{1}{2x}}}{x}$ is equal to

Limits & Continuity

$e$✓ Correct answer$0$$e-e^{2}$$\frac{-2}{e}$

Solution

$(1+2x)^{1/(2x)}=\exp\left(\frac{\ln(1+2x)}{2x}\right)$ and $\frac{\ln(1+2x)}{2x}=1-x+O(x^2)$. So $(1+2x)^{1/(2x)}=e(1-x+O(x^2))$ and $e-(\cdot)=ex+O(x^2)$. Limit $=e$. Option (A).

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