Limits, Continuity & Differentiability · JEE Main previous year questions

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function satisfying $f(x)+f(x+k)=n$, for all $x\in \mathbb{R}$ where $k>0$ and n is a positive integer. If $I_1={\int }_0^{4nk}f(x)dx$ and $I_2={\int }_{-k}^{3k}f(x)dx$, then :

1. A.$I_1+2I_2=4nk$
2. B.$I_1+2I_2=2nk$
3. C.$I_1+n{I}_2=4n^{2}k$
4. D.$I_1+n{I}_2=6n^{2}k$

The value of ${\lim }_{n\to \infty }6\tan \left\{{\sum }_{r=1}^n{\tan }^{-1}\left(\frac{1}{r^2+3r+3}\right)\right\}$ is equal to :

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

If ${\lim }_{x\to 1}\frac{\sin (3x^2-4x+1)-x^2+1}{2x^3-7x^2+ax+b}=-2$, then the value of $(a-b)$ is equal to ________.

Let $a$ be an integer such that ${\lim }_{x\to 7}\frac{18-[1-x]}{[x-3a]}$ exists, where $[t]$ is greatest integer $\le t$. Then $a$ is equal to :

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

Let $[t]$ denote the greatest integer $\le t$ and $\{t\}$ denote the fractional part of $t$. The integral value of $\alpha$ for which the left hand limit of the function $f(x)=[1+x]+\frac{{\alpha }^{2[x]+\{x\}}+[x]-1}{2[x]+\{x\}}$ at $x=0$ is equal to $\alpha -\frac{4}{3}$, is ________.

${\lim }_{x\to \frac{1}{\sqrt{2}}}\frac{\sin ({\cos }^{-1}x)-x}{1-\tan ({\cos }^{-1}x)}$ is equal to :

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

${\lim }_{x\to 0}\frac{\cos (\sin x)-\cos x}{x^4}$ is equal to :

1. A.$\frac{1}{3}$
2. B.$\frac{1}{4}$
3. C.$\frac{1}{6}$
4. D.$\frac{1}{12}$

Let $f(x)=\min \{1,1+x\sin x\}$, $0\le x\le 2\pi$. If m is the number of points, where f is not differentiable and n is the number of points, where f is not continuous, then the ordered pair (m, n) is equal to

1. A.$(2,0)$
2. B.$(1,0)$
3. C.$(1,1)$
4. D.$(2,1)$

Let $f(x)$ be a polynomial function such that $f(x)+f^{\prime }(x)+f^{\prime \prime }(x)=x^5+64$. Then, the value of $\underset{x\to 1}{\lim }\frac{f(x)}{x-1}$ is equal to:

1. A.$-15$
2. B.$-60$
3. C.$60$
4. D.$15$

Let $x=2t$, $y=\frac{t^2}{3}$ be a conic. Let $\mathrm{S}$ be the focus and $\mathrm{B}$ be the point on the axis of the conic such that $SA\perp BA$, where $\mathrm{A}$ is any point on the conic. If $\mathrm{k}$ is the ordinate of the centroid of the $\Delta \mathrm{SAB}$, then $\underset{t\to 1}{\lim }k$ is equal to :

1. A.$\frac{17}{18}$
2. B.$\frac{19}{18}$
3. C.$\frac{11}{18}$
4. D.$\frac{13}{18}$

$\underset{x\to \frac{\pi }{2}}{\lim }\left({\tan }^{2}x\left((2{\sin }^{2}x+3\sin x+4{)}^{\frac{1}{2}}-({\sin }^{2}x+6\sin x+2{)}^{\frac{1}{2}}\right)\right)$ is equal to

1. A.$\frac{1}{12}$
2. B.$-\frac{1}{18}$
3. C.$-\frac{1}{12}$
4. D.$\frac{1}{6}$

Let $f(x)=[2x^2+1]$ and $g(x)=\{\begin{array}{ll}2x-3,& x<0\\ 2x+3,& x\ge 0\end{array}$, where $[t]$ is the greatest integer $\le t$. Then, in the open interval $(-1,1)$, the number of points where $f\circ g$ is discontinuous is equal to $\underline{}$.

Let $$f(x)=\{\begin{array}{ll}\frac{\sin (x-[x])}{x-[x]}& ,x\in (-2,-1)\\ \max \{2x,3[|x|]\}& ,|x|<1\\ 1& ,otherwise\end{array}$$ where [t] denotes greatest integer $\le t$. If m is the number of points where $f$ is not continuous and n is the number of points where $f$ is not differentiable, then the ordered pair (m, n) is :

1. A.(3, 3)
2. B.(2, 4)
3. C.(2, 3)
4. D.(3, 4)

${\lim }_{n\to \infty }\left(\frac{n^2}{(n^2+1)(n+1)}+\frac{n^2}{(n^2+4)(n+2)}+\frac{n^2}{(n^2+9)(n+3)}+\cdots +\frac{n^2}{(n^2+n^2)(n+n)}\right)$ is equal to :

1. A.$\frac{\pi }{8}+\frac{1}{4}{\log }_{e}2$
2. B.$\frac{\pi }{4}+\frac{1}{8}{\log }_{e}2$
3. C.$\frac{\pi }{4}-\frac{1}{8}{\log }_{e}2$
4. D.$\frac{\pi }{8}+{\log }_e\sqrt{2}$

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function. Then ${\lim }_{x\to \frac{\pi }{4}}\frac{\frac{\pi }{4}{\int }_2^{{\sec }^{2}x}f(x)dx}{x^2-\frac{{\pi }^2}{16}}$ is equal to :

1. A.$f(2)$
2. B.$2f(2)$
3. C.$2f\left(\sqrt{2}\right)$
4. D.$4f(2)$

Let $f(x)=x^6+2x^4+x^3+2x+3$, $x\in \mathbb{R}$. Then the natural number n for which ${\lim }_{x\to 1}\frac{x^{n}f(1)-f(x)}{x-1}=44$ is ___________.

Let $[t]$ denote the greatest integer $\le t$. The number of points where the function $f(x)=[x]|x^2-1|+\sin \left(\frac{\pi }{[x]+3}\right)-[x+1]$, $x\in (-2,2)$ is not continuous is ___________.

${\lim }_{x\to 2}\left({\sum }_{n=1}^9\frac{x}{n(n+1)x^2+2(2n+1)x+4}\right)$ is equal to :

1. A.$\frac{9}{44}$
2. B.$\frac{5}{24}$
3. C.$\frac{1}{5}$
4. D.$\frac{7}{36}$

The value of $\underset{x\to 0}{\lim }\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to :

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

Let f : R $\to$ R be defined as $$f(x)=\{\begin{array}{ll}\frac{\lambda \left|x^2-5x+6\right|}{\mu (5x-x^2-6)},& x<2\\ e^{\frac{\tan (x-2)}{x-[x]}},& x>2\\ \mu ,& x=2\end{array}$$ where [x] is the greatest integer is than or equal to x. If f is continuous at x = 2, then $\lambda +\mu$ is equal to :

1. A.$e(-e+1)$
2. B.$e(e-2)$
3. C.1
4. D.$2e-1$

Consider the function $$f(x)=\{\begin{array}{ll}\frac{P(x)}{\sin (x-2)},& x\ne 2\\ 7,& x=2\end{array}$$ where P(x) is a polynomial such that P''(x) is always a constant and P(3) = 9. If f(x) is continuous at x = 2, then P(5) is equal to ____.

Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $$f(x)=\{\begin{array}{ll}\frac{x^3}{(1-\cos 2x{)}^2}{\log }_e\left(\frac{1+2x{e}^{-2x}}{(1-x{e}^{-x}{)}^2}\right),& x\ne 0\\ \alpha ,& x=0\end{array}$$ If f is continuous at $x=0$, then $\alpha$ is equal to :

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

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined as $$f(x)=\{\begin{array}{ll}3\left(1-\frac{|x|}{2}\right)& \text{if }|x|\le 2\\ 0& \text{if }|x|>2\end{array}$$ Let $g:\mathbb{R}\to \mathbb{R}$ be given by $g(x)=f(x+2)-f(x-2)$. If n and m denote the number of points in $\mathbb{R}$ where g is not continuous and not differentiable, respectively, then $n+m$ is equal to ________.

Let a function $f:\mathrm{R}\to \mathrm{R}$ be defined as $f(x)=\{\begin{array}{llc}\sin x-e^x& if& x\le 0\\ a+[-x]& if& 0<x<1\\ 2x-b& if& x\ge 1\end{array}$ where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:

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

If the value of ${\lim }_{x\to 0}(2-\cos x\sqrt{\cos 2x}{)}^{\left(\frac{x+2}{x^2}\right)}$ is equal to $e^a$, then a is equal to ________.