JEE Advanced · 2020 · 27 Sep 2020 · Paper 2 · Limits XII

More JEE Advanced questions on Limits XII

1. Let $\alpha$ and $\beta$ be the real numbers such that
   

   $$\underset{x\to 0}{\lim }\frac{1}{x^3}\left(\frac{\alpha }{2}{\int }_0^x\frac{1}{1-t^2}dt+\beta x\cos x\right)=2$$

   
   Then the value of $\alpha +\beta$ is ____.
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2. Let $k\in \mathbb{R}$. If ${\lim }_{x\to 0^{+}}(\sin (\sin kx)+\cos x+x{)}^{\frac{2}{x}}=e^6$, then the value of $k$ is
   JEE Advanced 2024 Copy linkView solution →
3. Let $\mathrm{f}:(0,1)\to \mathbb{R}$ be the functions defined as $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{n}}$ if $\mathrm{x}\in \left[\frac{1}{\mathrm{n}+1},\frac{1}{\mathrm{n}}\right)$ where $\mathrm{n}\in \mathrm{N}$. Let $\mathrm{g}:(0,1)\to \mathbb{R}$ be a …
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4. Let $\alpha$ be a positive real number. Let $f:\mathbb{R}\to \mathbb{R}$ and $g:(\alpha ,\infty )\to \mathbb{R}$ be the functions defined by …
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5. If
   

   $$\beta =\underset{x\to 0}{\lim }\frac{e^{x^3}-(1-x^3{)}^{\frac{1}{3}}+((1-x^2{)}^{\frac{1}{2}}-1)\sin x}{x{\sin }^{2}x}$$

   
   then the value of $6\beta$ is ____.
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6. Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit
   

   $$\underset{x\to 0^{+}}{\lim }\frac{(1-x{)}^{\frac{1}{x}}-e^{-1}}{x^a}$$

   
   is equal to a nonzero real number, is ____.
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