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Let $f:R\to \left( 0,\infty \right)$ be such that $f\left( x \right)+\frac{e^{x+x^{2}}}{f\left( x \right)}\le e^{x}+e^{x^{2}}, \forall$ $x>0$. Then, $\displaystyle\lim_{x \to 1} f\left( x \right)$ is

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1

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$\dfrac{1}{e}$

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e

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2e

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Mathematics

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MediumSingle Correct

Let $f:R\to \left( 0,\infty \right)$ be such that $f\left( x \right)+\frac{e^{x+x^{2}}}{f\left( x \right)}\le e^{x}+e^{x^{2}}, \forall$ $x>0$. Then, $\displaystyle\lim_{x \to 1} f\left( x \right)$ is

HardSingle Correct

The value of $\sum_{n=2}^{1947}\frac{1}{2^n+\sqrt{2^{1947}}}$ is equal to

MediumSingle Correct

If $\binom{30}{1}^{2}+2\binom{30}{2}^{2}+3\binom{30}{3}^{2}+...+30\binom{30}{30}^{2}=\frac{\alpha(60!)}{(30!)^{2}}$ then $\alpha$ is equal to

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