\(\lim_{x\to\pi} \dfrac{\sin x}{x-\pi}\) is
1
2
-1
-2
\(\lim_{x\to 0} \dfrac{x^{2}\cos x}{1-\cos x}\) is
2
\(\tfrac{3}{2}\)
\(-\tfrac{3}{2}\)
1
\(\lim_{x\to 0} \dfrac{(1+x)^{n}-1}{x}\) is
\(n\)
1
\(-n\)
0
\(\lim_{x\to 1} \dfrac{x^{m}-1}{x^{n}-1}\) is
1
\(\dfrac{m}{n}\)
\(-\dfrac{m}{n}\)
\(\dfrac{m^{2}}{n^{2}}\)
\(\lim_{\theta\to 0} \dfrac{1-\cos 4\theta}{1-\cos 6\theta}\) is
\(\dfrac{4}{9}\)
\(\dfrac{1}{2}\)
\(-\dfrac{1}{2}\)
-1
\(\lim_{x\to 0} \dfrac{\csc x - \cot x}{x}\) is
\(-\tfrac{1}{2}\)
1
\(\tfrac{1}{2}\)
1
\(\lim_{x\to 0} \dfrac{\sin x}{\sqrt{x+1}-\sqrt{1-x}}\) is
2
0
1
-1
\(\lim_{x\to \dfrac{\pi}{4}} \dfrac{\sec^{2}x -2}{\tan x -1}\) is
3
1
0
\(\sqrt{2}\)
\(\lim_{x\to 1} \dfrac{(\sqrt{x}-1)(2x-3)}{2x^{2}+x-3}\) is
\(\dfrac{1}{10}\)
\(-\dfrac{1}{10}\)
1
None of these
Let \(f(x)=\begin{cases}\dfrac{\sin[ x ]}{[ x ]},&[ x ]\ne 0\\0,&[ x ]=0\end{cases}\) where \([\cdot]\) denotes the greatest integer function. Then \(\lim_{x\to 0} f(x)\) is
1
0
-1
None of these
\(\lim_{x\to 0} \dfrac{|\sin x|}{x}\) is
1
-1
does not exist
None of these
Let \(f(x)=\begin{cases}x^{2}-1,&0
\(x^{2}-6x+9=0\)
\(x^{2}-7x+8=0\)
\(x^{2}-14x+49=0\)
\(x^{2}-10x+21=0\)
\(\lim_{x\to 0} \dfrac{\tan 2x - x}{3x - \sin x}\)
2
\(\dfrac{1}{2}\)
\(-\dfrac{1}{2}\)
\(\dfrac{1}{4}\)
Let \(f(x)=x-[x]\), \(x\in\mathbb{R}\). Then \(f'(\tfrac{1}{2})\) is
\(\dfrac{3}{2}\)
1
0
-1
If \(y=\sqrt{x}+\dfrac{1}{\sqrt{x}}\), then \(\dfrac{dy}{dx}\) at \(x=1\) is
1
\(\dfrac{1}{2}\)
\(\dfrac{1}{\sqrt{2}}\)
0
If \(f(x)=\dfrac{x-4}{2\sqrt{x}}\), then \(f'(1)\) is
\(\dfrac{5}{4}\)
\(\dfrac{4}{5}\)
1
0
If \(y=\dfrac{1+\dfrac{1}{x^{2}}}{1-\dfrac{1}{x^{2}}}\), then \(\dfrac{dy}{dx}\) is
\(-\dfrac{4x}{(x^{2}-1)^{2}}\)
\(-\dfrac{4x}{x^{2}-1}\)
\(\dfrac{1-x^{2}}{4x}\)
\(\dfrac{4x}{x^{2}-1}\)
If \(y=\dfrac{\sin x+\cos x}{\sin x-\cos x}\), then \(\dfrac{dy}{dx}\) at \(x=0\) is
-2
0
\(\dfrac{1}{2}\)
does not exist
If \(y=\dfrac{\sin(x+9)}{\cos x}\), then \(\dfrac{dy}{dx}\) at \(x=0\) is
\(\cos 9\)
\(\sin 9\)
0
1
If \(f(x)=1+x+\dfrac{x^{2}}{2}+\dots+\dfrac{x^{100}}{100}\), then \(f'(1)\) is equal to
\(\dfrac{1}{100}\)
100
does not exist
0
If \(f(x)=\dfrac{x^{n}-a^{n}}{x-a}\) for some constant \(a\), then \(f'(a)\) is
1
0
does not exist
\(\dfrac{1}{2}\)
If \(f(x)=x^{100}+x^{99}+\dots+x+1\), then \(f'(1)\) is equal to
5050
5049
5051
50051
If \(f(x)=1-x+x^{2}-x^{3}+\dots+ x^{100}\), then \(f'(1)\) is equal to
150
-50
-150
50