JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Limits

More JEE Main questions on Limits

1. Let $f:\mathrm{R}\to (0,\infty )$ be a twice differentiable function such that $\mathrm{f}(3)=18,{\mathrm{f}}^{\prime }(3)=0$ and ${\mathrm{f}}^{\prime \prime }(3)=4$.
   Then …
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2. If ${\lim }_{x\to 0}\frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}}{x\cos x-{\log }_e(1+x)}=2$, then $a^2+b^2+c^2$ is equal to :
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3. The value of
   ${\lim }_{x\to 0}\frac{{\log }_e\left(\sec (ex)\cdot \sec \left(e^{2}x\right)\cdot \dots \cdot \sec \left(e^{10}x\right)\right)}{e^2-e^{2\cos x}}$
   is equal to
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4. If ${\lim }_{x\to 2}\frac{\sin (x^3-5x^2+ax+b)}{(\sqrt{x-1}-1){\log }_e(x-1)}=m$, then $a+b+m$ is equal to:
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5. The product of all possible values of $\alpha$, for which ${\lim }_{x\to 0}\left(\frac{1-\cos (\alpha x)\cos ((\alpha +1)x)\cos ((\alpha +2)x)}{{\sin }^2((\alpha +1)x)}\right)=2$, is :
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6. Let $f(x)={\lim }_{y\to 0}\frac{(1-\cos (xy))\tan (xy)}{y^3}$. Then the number of solutions of the equation $f(\mathrm{x})=\sin \mathrm{x}$, $x\in R$ is:
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