Chapter 1: Problem 26
The value of \(\lim _{x \rightarrow \infty}\left(x^{4}(\ln x)^{16}\right)\) is (A) (B) 0 (C) \(\frac{1}{2}\) (D) \(-\frac{1}{2}\)
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Column - I (A) $\lim _{n \rightarrow \infty} \cos ^{2}\left(\pi\left(\sqrt[3]{n^{3}+n^{2}+2 n}\right)\right)\( where \)n$ is an integer, equals. (B) $\lim _{n \rightarrow \infty} \mathrm{n} \sin \left(2 \pi \sqrt{1+\mathrm{n}^{2}}\right)(\mathrm{n} \in \mathrm{N})$ equals. (C) $\lim _{n \rightarrow \infty}(-1)^{n} \sin \left(\pi \sqrt{n^{2}+0.5 n+1}\right)\left(\sin \frac{(n+1) \pi}{2 n}\right)\( is (where \)\left.n \in N\right)$. (D) If \(\lim _{x \rightarrow \infty}\left(\frac{x+a}{x-a}\right)^{x}=e\) where 'a' is some real constant then the value of 'a' is equal to. Column - II (P) \(\frac{1}{\sqrt{2}}\) (Q) \(\frac{1}{4}\) (R) \(\pi\) (S) non existent
The function(s) which have a limit as \(\mathrm{n} \rightarrow \infty\) (A) \(\left(\frac{n-1}{n+1}\right)^{2}\) (B) \((-1)^{n}\left(\frac{n-1}{n+1}\right)^{2}\) (C) \(\frac{n^{2}+1}{n}\) (D) \((-1)^{n} \frac{n^{2}+1}{n}\)
\(\mathrm{A}_{0}\) is an equilateral triangle of unit area, \(\mathrm{A}_{0}\) is divided into four equal parts, each an equilateral triangle, by joining the mid points of the sides of \(\mathrm{A}_{0}\). The central triangle is removed. Treating the remaining three triangles in the same way of division as was done to \(\mathrm{A}_{0}\), and this process is repeated \(\mathrm{n}\) times. The sum of the area of the triangles removed in \(\mathrm{S}_{\mathrm{n}}\) then $\lim _{\mathrm{n} \rightarrow \infty} \mathrm{S}_{\mathrm{n}}$ is (A) \(1 / 2\) (B) 1 (C) \(-1\) (D) 2
Which of the following functions have a graph which lies between the graphs of \(\mathrm{y}=|\mathrm{x}|\) and \(\mathrm{y}=-|\mathrm{x}|\) and have a limiting value as \(\mathrm{x} \rightarrow 0\). (A) \(\mathrm{y}=\mathrm{x} \cos \mathrm{x}\) (B) \(y=|x| \sin x\) (C) \(\mathrm{y}=\mathrm{x} \cos \frac{\mathrm{l}}{\mathrm{x}}\) (D) \(\mathrm{y}=\left|\mathrm{x} \sin \frac{1}{\mathrm{x}}\right|\)
Consider the function \(f(x)=\left(\frac{a x+1}{b x+2}\right)^{x}\) where \(a^{2}+b^{2} \neq 0\) then \(\lim f(x)\) (A) exists for all values of \(a\) and \(b\) (B) is zero for \(\mathrm{a}<\mathrm{b}\) (C) is non existent for \(\mathrm{a}>\mathrm{b}\) (D) is e \({ }^{-\left(\frac{1}{a}\right)}\) or \(e^{-\left(\frac{1}{b}\right)}\) if \(a=b\)
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