JEE Advanced · 2024 · 26 May 2024 · 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 ____.
   JEE Advanced 2025 Copy linkView solution →
2. 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 …
   JEE Advanced 2023 Copy linkView solution →
3. 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 …
   JEE Advanced 2022 Copy linkView solution →
4. 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 ____.
   JEE Advanced 2022 Copy linkView solution →
5. 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 ____.
   JEE Advanced 2020 Copy linkView solution →
6. The value of the limit
   

   $$\underset{x\to \frac{\pi }{2}}{\lim }\frac{4\sqrt{2}(\sin 3x+\sin x)}{\left(2\sin 2x\sin \frac{3x}{2}+\cos \frac{5x}{2}\right)-\left(\sqrt{2}+\sqrt{2}\cos 2x+\cos \frac{3x}{2}\right)}$$

   
   is ____
   JEE Advanced 2020 Copy linkView solution →