If $\mathop {\lim }\limits_{n \to \infty } \frac{1}{{10 + {{\left( {2\cos x} \right)}^{2n}}}} = 0$,then the complete set of all possible values of $|\sin x|$ is:
- A$[0, \frac{\sqrt{3}}{2})$
- B$[\frac{\sqrt{3}}{2}, 1]$
- C$[\frac{1}{2}, 1]$
- D$[\frac{1}{2}, \frac{\sqrt{3}}{2}]$
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If $\lim_{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(nk+1)+(nk+2)+\ldots+(nk+n)] = 33 \cdot \lim_{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot [1^k + 2^k + 3^k + \ldots + n^k]$,then the integral value of $k$ is equal to $....$
DifficultJEE MAIN 2022
View SolutionLet $[.]$ denote the greatest integer function. Assertion $(A) : \lim_{x \rightarrow \infty} \frac{[x]}{x} = 1$. Reason $(R) : f(x) = x - 1, g(x) = [x], h(x) = x$ and $\lim_{x \rightarrow \infty} \frac{f(x)}{x} = \lim_{x \rightarrow \infty} \frac{h(x)}{x} = 1$.
If $L = \lim_{x^2 \to a} \frac{b - \cos(x^2 - a)}{(x^2 - a) \sin(c(x^2 - a))}$ is a non-zero finite value $(a > 0)$,then:
If $\lim_{x \to 0} \frac{(4^x - 1)^3}{\tan(\frac{x}{4}) \log(1 + \frac{x^2}{3})} = 96(\log a)^b$, then $(a + b) = $
DifficultMHT CET 2026
View Solution$\lim _{x \rightarrow 0}\left(\frac{\sinh 2 x}{2 x}\right)^{\frac{1}{x^2}} = $
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