Question

The value of $\underset{n\to \infty }{\lim }\frac{\left[r\right]+\left[2r\right]+...+\left[nr\right]}{n^2},$ where $r$ is non-zero real number and $\left[r\right]$ denotes the greatest integer less than or equal to $r,$ is equal to :

JEE Main 2021 (17 Mar Shift 2)

Options

• A: $\frac{r}{2}$
• B: $r$
• C: $2r$
• D: $0$

Question

$\underset{x\to 2}{\lim }\left(\sum _{n=1}^9\frac{x}{n(n+1)x^2+2(2n+1)x+4}\right)$ is equal to :

JEE Main 2021 (26 Aug Shift 2)

Options

• A: $\frac{5}{24}$
• B: $\frac{7}{36}$
• C: $\frac{1}{5}$
• D: $\frac{9}{44}$

Question

JEE Main 2021 (26 Feb Shift 1)

Options

• A: $\frac{4}{3}$
• B: $\frac{2}{\sqrt{3}}$
• C:

  $\frac{2}{3}$

• D: $\frac{3}{4}$

Question

If $\alpha , \beta$ are the distinct roots of $x^2+bx+c=0,$ then $\underset{x\to \beta }{\lim }\frac{e^{2\left(x^2+bx+c\right)}-1-2\left(x^2+bx+c\right)}{(x-\beta {)}^2}$ is equal to

JEE Main 2021 (27 Aug Shift 1)

Options

• A: $2\left(b^2+4c\right)$
• B: $b^2-4c$
• C: $2\left(b^2-4c\right)$
• D: $b^2+4c$

Question

The value of $\underset{x\to 0}{\lim }\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to :

JEE Main 2021 (27 Jul Shift 2)

Options

• A: $0$
• B: $4$
• C: $-4$
• D: $-1$

Question

If $\underset{x\to 0}{\lim }\left[\frac{\alpha x{e}^x-\beta {\log }_e(1+x)+\gamma x^2{\mathrm{e}}^{-x}}{x{\sin }^{2}x}\right]=10, \alpha , \beta , \gamma \in R$, then the value of $\alpha +\beta +\gamma$ is __________.

JEE Main 2021 (20 Jul Shift 2)

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Question

If $\underset{n\to \infty }{\lim }\left(\sqrt{n^2-n-1}+n\alpha +\beta \right)=0$ then $8\left(\alpha +\beta \right)$ is equal to

JEE Main 2022 (25 Jul Shift 1)

Options

• A: $4$
• B: $-8$
• C: $-4$
• D: $8$

Question

Let $\beta =\underset{x\to 0}{\lim }\frac{\alpha x-\left(e^{3x}-1\right)}{\alpha x\left(e^{3x}-1\right)}$ for some $\alpha \in \mathrm{ℝ}$. Then the value of $\alpha +\beta$ is:

JEE Main 2022 (26 Jul Shift 2)

Options

• A: $\frac{14}{5}$
• B: $\frac{3}{2}$
• C: $\frac{5}{2}$
• D: $\frac{7}{2}$

Question

$\underset{x\to 0}{\lim }\frac{\cos \left(\sin x\right)-\cos x}{x^4}$ is equal to

JEE Main 2022 (26 Jun Shift 2)

Options

• A: $\frac{1}{3}$
• B: $\frac{1}{6}$
• C: $\frac{1}{4}$
• D: $\frac{1}{12}$

Question

If $\underset{x\to 0}{\lim }\frac{\alpha e^x+\beta e^{-x}+\gamma \sin x}{x{\sin }^{2}x}=\frac{2}{3}$, where $\alpha ,\beta ,\gamma \in R$, then which of the following is NOT correct?

JEE Main 2022 (29 Jul Shift 1)

Options

• A: ${\alpha }^2+{\beta }^2+{\gamma }^2=6$
• B: $\alpha \beta +\beta \gamma +\gamma \alpha +1=0$
• C: $\alpha {\beta }^2+\beta {\gamma }^2+\gamma {\alpha }^2+3=0$
• D: ${\alpha }^2-{\beta }^2+{\gamma }^2=4$

Question

$\underset{x\to \infty }{\lim }\frac{{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)}^6+{\left(\sqrt{3x+1}-\sqrt{3x-1}\right)}^6}{{\left(x+\sqrt{x^2-1}\right)}^6+{\left(x-\sqrt{x^2-1}\right)}^6}{x}^3$

JEE Main 2023 (31 Jan Shift 2)

Options

• A: is equal to $\frac{27}{2}$
• B: is equal to $9$
• C: does not exist
• D: is equal to $27$

Question

$\underset{n\to \infty }{\lim }\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right)....\left(2^{\frac{1}{2}}-2^{\frac{1}{2n+1}}\right)\right\}$ is equal to

JEE Main 2023 (06 Apr Shift 2)

Options

• A: $1$
• B: $0$
• C: $\sqrt{2}$
• D: $\frac{1}{\sqrt{2}}$

Question

If $\underset{x\to 0}{\lim }\frac{e^{ax}-\cos \left(bx\right)-\frac{cx{e}^{-cx}}{2}}{1-\cos \left(2x\right)}=17$, then $5a^2+b^2$ is equal to

JEE Main 2023 (13 Apr Shift 2)

Options

• A: $64$
• B: $72$
• C: $68$
• D: $76$

Question

If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $8 \mathrm{~m}+12 \mathrm{n}$ is equal to______

JEE Main 2024 (04 Apr Shift 1)

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Question

Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then $2 f(2)+3 f(3)$ is equal to _______

JEE Main 2024 (05 Apr Shift 1)

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Question

Let $f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a$ is equal to

JEE Main 2024 (06 Apr Shift 1)

Options

• A: $\frac{3}{2}+\frac{\pi}{4}$
• B: $\frac{3}{4}+\frac{\pi}{8}$
• C: $\frac{3}{8}+\frac{\pi}{4}$
• D: $\frac{5}{2}+\frac{\pi}{8}$

Question

$\lim _{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)}$ is equal to :

JEE Main 2024 (06 Apr Shift 2)

Options

• A: $\frac{2}{3}$
• B: $\frac{1}{3}$
• C: $\frac{3}{4}$
• D: $\frac{1}{2}$

Question

The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots \sqrt[10]{\cos 10 x}}{x^2}\right)$ is

JEE Main 2024 (08 Apr Shift 1)

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Question

Let $\lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.$ $\left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right)$ be $\frac{\pi}{k}$, using only the principal values of the inverse trigonometric functions. Then $\mathrm{k}^2$ is equal to ________

JEE Main 2024 (09 Apr Shift 1)

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Question

Let $f\left(x\right)=\left\{\begin{array}{l}x-1, x \text{is even},\\ 2x, x \text{is odd},\end{array}\right.x\in N$. If for some $a\in N,f\left(f\left(f\left(a\right)\right)\right)=21$, then $\underset{x\to a^{-}}{\lim }\left\{\frac{{\left|x\right|}^3}{a}-\left[\frac{x}{a}\right]\right\},$ where $\left[t\right]$ denotes the greatest integer less than or equal to $t$, is equal to:

JEE Main 2024 (01 Feb Shift 2)

Options

• A: $121$
• B: $144$
• C: $169$
• D: $225$

Question

$\underset{x\to 0}{\lim }\frac{e^{2\left|\sin x\right|}-2\left|\sin x\right|-1}{x^2}$

JEE Main 2024 (31 Jan Shift 1)

Options

• A: is equal to $-1$
• B: does not exist
• C: is equal to $1$
• D: is equal to $2$

Question

Let the slope of the line $45x+5y+3=0$ be $27r_1+\frac{9r_2}{2}$ for some $r_1, r_2\in R$. Then $\underset{x\to 3}{\lim }\left({\int }_3^x\frac{8t^2}{\frac{3r_{2}x}{2}-r_2{x}^2-r_1{x}^3-3x}dt\right)$ is equal to ______.

JEE Main 2024 (29 Jan Shift 2)

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