The value of limx→0 ex−cos⁡2x−xx2, is

# The value of $\underset{x\to 0}{lim} \frac{{e}^{x}-\mathrm{cos}2x-x}{{x}^{2}},$ is

1. A

2

2. B

$\frac{1}{2}$

3. C

$\frac{3}{2}$

4. D

$\frac{5}{2}$

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### Solution:

$\underset{x\to 0}{lim} \frac{{e}^{x}-\mathrm{cos}2x-x}{{x}^{2}}$

$\begin{array}{l}=\underset{x\to 0}{lim} \frac{\left({e}^{x}-1-x\right)+\left(1-\mathrm{cos}2x\right)}{{x}^{2}}\\ =\underset{x\to 0}{lim} \frac{\left(\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\dots \right)+2{\mathrm{sin}}^{2}x}{{x}^{2}}\\ =\underset{x\to 0}{lim} \left\{\left(\frac{1}{2!}+\frac{x}{3!}+\dots \right)+2{\left(\frac{\mathrm{sin}x}{x}\right)}^{2}\right\}=\frac{1}{2!}+2×{1}^{2}=\frac{5}{2}\end{array}$

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