Limits, Continuity & Differentiability · JEE Main previous year questions

Let $f(x)=\{\begin{array}{ll}\frac{1}{3},& x\le \pi /2\\ \frac{b(1-\sin x)}{(\pi -2x{)}^2},& x>\pi /2\end{array}$. If $f$ is continuous at $x=\pi /2$, then the value of ${\int }_0^{3b-6}|x^2+2x-3|dx$ is :

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

The value of ${\lim }_{x\to 0}\left(\frac{x^2{\sin }^{2}x}{x^2-{\sin }^{2}x}\right)$ is:

1. A.2
2. B.3
3. C.4
4. D.6

Let ${\lim }_{x\to 2}\frac{(\tan (x-2))(r{x}^2+(p-2)x-2p)}{(x-2{)}^2}=5$ for some $r,p\in \mathbf{R}$. If the set of all possible values of $q$, such that the roots of the equation $r{x}^2-px+q=0$ lie in $(0,2)$, be the interval $(\alpha ,\beta ]$, then $4(\alpha +\beta )$ equals :

1. A.$11$
2. B.$13$
3. C.$17$
4. D.$21$

The product of all possible values of $\alpha$, for which $$\underset{x\to 0}{\lim }\left(\frac{1-\cos (\alpha x)\cos ((\alpha +1)x)\cos ((\alpha +2)x)}{{\sin }^2((\alpha +1)x)}\right)=2,$$ is:

1. A.-2
2. B.1
3. C.-1
4. D.$\frac{5}{4}$

Let $f(x)={\lim }_{y\to 0}\frac{(1-\cos (xy))\tan (xy)}{y^3}$. Then the number of solutions of the equation $f(x)=\sin x$, $x\in \mathbb{R}$ is :

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

Let $f(x)=\{\begin{array}{ll}{e}^{x-1}& ,x<0\\ x^2-5x+6& ,x\ge 0\end{array}$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha +\beta$ is equal to ____.

If ${\lim }_{x\to 2}\frac{\sin \left(x^3-5x^2+ax+b\right)}{(\sqrt{x-1}-1){\log }_{\mathrm{e}}(x-1)}=\mathrm{m}$, then $\mathrm{a}+\mathrm{b}+\mathrm{m}$ is equal to :

1. A.5
2. B.6
3. C.8
4. D.10

Let $f(x)$ be a polynomial of degree 5, and have extrema at $x=1$ and $x=-1$. If ${\lim }_{x\to 0}\left(\frac{f(x)}{x^3}\right)=-5$, then $f(2)-f(-2)$ is equal to :

1. A.0
2. B.50
3. C.92
4. D.112

The number of points in the interval $[2,4]$, at which the function $f(x)=\left[x^2-x-\frac{1}{2}\right]$, where $[\cdot ]$ denotes the greatest integer function, is discontinuous, is ____.

The value of $$\underset{x\to 0}{\lim }\frac{{\log }_e(\sec (ex)\cdot \sec (e^{2}x)\cdot \cdots \cdot \sec (e^{10}x))}{e^2-e^{2\cos x}}$$ is equal to

1. A.$\frac{(e^{10}-1)}{2e^2(e^2-1)}$
2. B.$\frac{(e^{20}-1)}{2e^2(e^2-1)}$
3. C.$\frac{(e^{10}-1)}{2(e^2-1)}$
4. D.$\frac{(e^{20}-1)}{2(e^2-1)}$

Let $f(x)={\lim }_{\theta \to 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta }\right)}\sin (x-1)}{1+x^{\left(\frac{2}{\theta }\right)}(x-1)}\right),x\in \mathbb{R}$. Consider the following two statements : (I) $f(x)$ is discontinuous at $x=1$. (II) $f(x)$ is continuous at $x=-1$. Then,

1. A.Neither (I) nor (II) is True
2. B.Only (II) is True
3. C.Only (I) is True
4. D.Both (I) and (II) are True

Let $\alpha ,\beta \in \mathbb{R}$ be such that the function $f(x)=\{\begin{array}{ll}2\alpha (x^2-2)+2\beta x& ,x<1\\ (\alpha +3)x+(\alpha -\beta )& ,x\ge 1\end{array}$ be differentiable at all $x\in \mathbb{R}$. Then $34(\alpha +\beta )$ is equal to

1. A.48
2. B.84
3. C.36
4. D.24

If the function $f(x)=\frac{e^x\left(e^{\tan x-x}-1\right)+{\log }_e(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to

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

Let $f(x)=\{\begin{array}{ll}\frac{a{x}^2+2ax+3}{4x^2+4x-3}& ,x\ne -\frac{3}{2},\frac{1}{2}\\ b& ,x=-\frac{3}{2},\frac{1}{2}\end{array}$ be continuous at $x=-\frac{3}{2}$. If $f\circ f(x)=\frac{7}{5}$, then $x$ is equal to:

1. A.1.4
2. B.2
3. C.1
4. D.0

If $f(x)=\{\begin{array}{ll}\frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x}& ,x\ne 0\\ b& ,x=0\end{array}$ is continuous at $x=0$, then $a+b$ is equal to :

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

Let $f:\mathbf{R}\to (0,\infty )$ be a twice differentiable function such that $f(3)=18$, $f^{\prime }(3)=0$ and $f^{\prime \prime }(3)=4$. Then ${\lim }_{x\to 1}{\left({\log }_e\left(\frac{f(2+x)}{f(3)}\right)\right)}^{\frac{18}{(x-1{)}^2}}$ is equal to :

1. A.9
2. B.18
3. C.1
4. D.2

Given below are two statements: Statement I: ${\lim }_{x\to 0}\left(\frac{{\tan }^{-1}x+{\log }_e\sqrt{\frac{1+x}{1-x}}-2x}{x^5}\right)=\frac{2}{5}$ Statement II: ${\lim }_{x\to 1}\left(x^{\frac{2}{1-x}}\right)=\frac{1}{e^2}$ In the light of the above statements, choose the correct answer from the options given below:

1. A.Statement I is false but Statement II is true
2. B.Both Statement I and Statement II are false
3. C.Both Statement I and Statement II are true
4. D.Statement I is true but Statement II is false

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2x)-f(x)=x$ for all $x\in \mathbb{R}$. If ${\lim }_{n\to \infty }\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)$, then ${\sum }_{r=1}^{10}G\left(r^2\right)$ is equal to

1. A.$215$
2. B.$420$
3. C.$385$
4. D.$540$

If ${\lim }_{x\to 1^{+}}\frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1{)}^3}=-1$, where $\lambda ,\mu \in \mathbb{R}$, then $\lambda +\mu$ is equal to

1. A.$20$
2. B.$19$
3. C.$18$
4. D.$17$

Let $m$ and $n$ be the number of points at which the function $f(x)=\max \left\{x,x^3,x^5,\dots x^{21}\right\},x\in \mathbb{R}$, is not differentiable and not continuous, respectively. Then $m+n$ is equal to _________.

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1,f^{\prime }(2)=4$. Let ${\lim }_{x\to 0}(f(2+x){)}^{3/x}={\mathrm{e}}^{\alpha }$. Then the number of times the curve $y=4x^3-4x^2-4(\alpha -7)x-\alpha$ meets $x$-axis is :

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

Let $$f(x)=\{\begin{array}{ll}(1+ax{)}^{1/x}& ,x<0\\ 1+b,& x=0\\ \frac{(x+4{)}^{1/2}-2}{(x+c{)}^{1/3}-2},& x>0\end{array}$$ be continuous at $x=0$. Then $e^{a}bc$ is equal to:

1. A.64
2. B.48
3. C.36
4. D.72

If ${\lim }_{x\to 0}{\left(\frac{\tan x}{x}\right)}^{\frac{1}{x^2}}=p$, then $96{\log }_{\mathrm{e}}p$ is equal to _________

For $\alpha ,\beta ,\gamma \in \mathbf{R}$, if ${\lim }_{x\to 0}\frac{x^2\sin \alpha x+(\gamma -1)e^{x^2}}{\sin 2x-\beta x}=3$, then $\beta +\gamma -\alpha$ is equal to :

1. A.-1
2. B.4
3. C.6
4. D.7

If ${\lim }_{x\to 0}\frac{\cos (2x)+a\cos (4x)-b}{x^4}$ is finite, then $(a+b)$ is equal to:

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