Limits, Continuity & Differentiability · JEE Main previous year questions

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

1. A.$\sqrt{2}$
2. B.1
3. C.$\frac{1}{\sqrt{2}}$
4. D.0

Let $f(x)=\frac{x}{(1+x^n{)}^{\frac{1}{n}}},x\in \mathbb{R}-\{-1\},n\in \mathbb{N},n>2$. If $f^n(x)=(f\circ f\circ f\dots \text{ upto }n\text{ times})(x)$, then ${\lim }_{n\to \infty }{\int }_0^1{x}^{n-2}\left(f^n(x)\right)\mathrm{d}x$ is equal to ____.

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

1. A.is equal to 9
2. B.is equal to $\frac{27}{2}$
3. C.does not exist
4. D.is equal to 27

${\lim }_{x\to 0}\frac{48}{x^4}{\int }_0^x\frac{t^3}{t^6+1}\mathrm{d}t$ is equal to ____.

Let $y=f(x)$ be the solution of the differential equation $y(x+1)dx-x^{2}dy=0,y(1)=e$. Then $\underset{x\to 0^{+}}{\lim }f(x)$ is equal to

1. A.$e^2$
2. B.0
3. C.$\frac{1}{e^2}$
4. D.$\frac{1}{e}$

Let $x=2$ be a root of the equation $x^2+px+q=0$ and $f(x)=\{\begin{array}{ll}\frac{1-\cos (x^2-4px+q^2+8q+16)}{(x-2p{)}^4},& x\ne 2p\\ 0,& x=2p\end{array}$ Then $\underset{x\to 2p^{+}}{\lim }[f(x)]$, where $[.]$ denotes greatest integer function, is

1. A.2
2. B.1
3. C.0
4. D.$-1$

The value of $\underset{n\to \infty }{\lim }\frac{1+2-3+4+5-6+.....+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}}$ is :

1. A.$\frac{3}{2\sqrt{2}}$
2. B.$\frac{3}{2}(\sqrt{2}+1)$
3. C.$3(\sqrt{2}+1)$
4. D.$\frac{\sqrt{2}+1}{2}$

If the function $f(x)=\{\begin{array}{ll}(1+|\cos x|{)}^{\frac{\lambda }{|\cos x|}},& 0<x<\frac{\pi }{2}\\ \mu ,& x=\frac{\pi }{2}\\ e^{\frac{\cot 6x}{\cot 4x}},& \frac{\pi }{2}<x<\pi \end{array}$ is continuous at $x=\frac{\pi }{2}$, then $9\lambda +6{\log }_e\mu +{\mu }^6-e^{6\lambda }$ is equal to

1. A.11
2. B.10
3. C.8
4. D.$2e^4+8$

Let $y=y(t)$ be a solution of the differential equation $\frac{dy}{dt}+\alpha y=\gamma e^{-\beta t}$ where, $\alpha >0,\beta >0$ and $\gamma >0$. Then $\underset{t\to \infty }{\lim }y(t)$

1. A.is 0
2. B.is 1
3. C.is $-1$
4. D.does not exist

${\lim }_{t\to 0}{\left(1^{\frac{1}{{\sin }^{2}t}}+2^{\frac{1}{{\sin }^{2}t}}+\dots +n^{\frac{1}{{\sin }^{2}t}}\right)}^{{\sin }^{2}t}$ is equal to

1. A.$\frac{n(n+1)}{2}$
2. B.$n$
3. C.$n^2+n$
4. D.$n^2$

The set of all values of $a$ for which $\underset{x\to a}{\lim }([x-5]-[2x+2])=0$, where $[\alpha ]$ denotes the greatest integer less than or equal to $\alpha$ is equal to

1. A.$[-7.5,-6.5]$
2. B.$(-7.5,-6.5]$
3. C.$[-7.5,-6.5)$
4. D.$(-7.5,-6.5)$

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

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

Let the function $f(x)=\{\begin{array}{ll}\frac{{\log }_e(1+5x)-{\log }_e(1+\alpha x)}{x}& ;\text{if }x\ne 0\\ 10& ;\text{if }x=0\end{array}$ be continuous at $x=0$. Then $\alpha$ is equal to

1. A.10
2. B.$-10$
3. C.5
4. D.$-5$

If $[t]$ denotes the greatest integer $\le t$, then the number of points, at which the function $f(x)=4|2x+3|+9\left[x+\frac{1}{2}\right]-12[x+20]$ is not differentiable in the open interval $(-20,20)$, is ________.

$\underset{x\to 0}{\lim }{\left(\frac{(x+2\cos x{)}^3+2(x+2\cos x{)}^2+3\sin (x+2\cos x)}{(x+2{)}^3+2(x+2{)}^2+3\sin (x+2)}\right)}^{\frac{100}{x}}$ is equal to ____.

The function $f:\mathbb{R}\to \mathbb{R}$ defined by $$f(x)=\underset{n\to \infty }{\lim }\frac{\cos (2\pi x)-x^{2n}\sin (x-1)}{1+x^{2n+1}-x^{2n}}$$ is continuous for all x in :

1. A.$\mathbb{R}-\{-1\}$
2. B.$\mathbb{R}-\{-1,1\}$
3. C.$\mathbb{R}-\{1\}$
4. D.$\mathbb{R}-\{0\}$

Let a function $f:\mathbb{R}\to \mathbb{R}$ be defined as : $$f(x)=\{\begin{array}{ll}{\int }_0^x(5-|t-3|)dt,& x>4\\ x^2+bx,& x\le 4\end{array}$$ where $b\in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

1. A.$f$ is not differentiable at $x=4$
2. B.$f^{\prime }(3)+f^{\prime }(5)=\frac{35}{4}$
3. C.$f$ is increasing in $(-\infty ,\frac{1}{8})\cup (8,\infty )$
4. D.$f$ has a local minima at $x=\frac{1}{8}$

If for $p\ne q\ne 0$, the function $f(x)=\frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then :

1. A.$7pqf(0)-1=0$
2. B.$63qf(0)-p^2=0$
3. C.$21qf(0)-p^2=0$
4. D.$7pqf(0)-9=0$

Let $f:\mathrm{R}\to \mathrm{R}$ be a continuous function such that $f(3x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to :

1. A.4
2. B.10
3. C.11
4. D.16

If the function $f(x)=\{\begin{array}{ll}\frac{{\log }_{\mathrm{e}}(1-x+x^2)+{\log }_{\mathrm{e}}(1+x+x^2)}{\sec x-\cos x}& ,x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)-\{0\}\\ k& ,x=0\end{array}$ is continuous at $x=0$, then k is equal to:

1. A.1
2. B.$-1$
3. C.e
4. D.0

If $f(x)=\{\begin{array}{ll}x+a& ,x\le 0\\ |x-4|& ,x>0\end{array}$ and $g(x)=\{\begin{array}{ll}x+1& ,x<0\\ (x-4{)}^2+b& ,x\ge 0\end{array}$ are continuous on R, then $(gof)(2)+(fog)(-2)$ is equal to :

1. A.$-10$
2. B.10
3. C.8
4. D.$-8$

${\lim }_{x\to \frac{\pi }{4}}\frac{8\sqrt{2}-(\cos x+\sin x{)}^7}{\sqrt{2}-\sqrt{2}\sin 2x}$ is equal to

1. A.14
2. B.7
3. C.$14\sqrt{2}$
4. D.$7\sqrt{2}$

${\lim }_{n\to \infty }{\sum }_{r=1}^n\frac{r}{2r^2-7rn+6n^2}$ is equal to :

1. A.${\log }_{\mathrm{e}}\left(\frac{\sqrt{3}}{2}\right)$
2. B.${\log }_{\mathrm{e}}\left(\frac{3\sqrt{3}}{4}\right)$
3. C.${\log }_{\mathrm{e}}\left(\frac{27}{4}\right)$
4. D.${\log }_{\mathrm{e}}\left(\frac{4}{3}\right)$

Suppose ${\lim }_{x\to 0}\frac{F(x)}{x^3}$ exists and is equal to L, where $$F(x)=\left|\begin{array}{ccc}a+\sin \frac{x}{2}& -b\cos x& 0\\ -b\cos x& 0& a+\sin \frac{x}{2}\\ 0& a+\sin \frac{x}{2}& -b\cos x\end{array}\right|.$$ Then, $-112\mathrm{L}$ is equal to ____ .

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f\left(\frac{\pi }{4}\right)=\sqrt{2}$, $f\left(\frac{\pi }{2}\right)=0$ and $f^{\prime }\left(\frac{\pi }{2}\right)=1$ and let $g(x)={\int }_x^{\pi /4}(f^{\prime }(t)\sec t+\tan t\sec tf(t))dt$ for $x\in \left[\frac{\pi }{4},\frac{\pi }{2}\right)$. Then ${\lim }_{x\to {\left(\frac{\pi }{2}\right)}^{-}}g(x)$ is equal to :

1. A.$2$
2. B.$3$
3. C.$4$
4. D.$-3$