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Yukti Vault

Calculus

Limits & Continuity: JEE Main & Advanced PYQs

282 previous year questions on Limits & Continuity from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 282

  1. If $f:\mathbf{R}\to\mathbf{R}$ is a function defined by $f(x)=[x-1]\cos\left(\frac{2x-1}{2}\right)\pi$, where $[\cdot]$ denotes the greatest integer function, then $f$ …
  2. Let $\lim_{x\to2}\frac{(\tan(x-2))\left(rx^{2}+(p-2)x-2p\right)}{(x-2)^{2}}=5$ for some $r,p\in\mathbf{R}$. If the set of all possible values of $q$, such that the roots …
  3. Let $f(x)=\lim_{y\to0}\frac{(1-\cos(xy))\tan(xy)}{y^{3}}$. Then the number of solutions of the equation $f(x)=\sin x$, $x\in\mathbf{R}$ is :
  4. Let $f(x)=\begin{cases}\frac{1}{3}, & x\le\frac{\pi}{2}\\ \frac{b(1-\sin x)}{(\pi-2x)^{2}}, & x>\frac{\pi}{2}\end{cases}$. If $f$ is continuous at $x=\pi/2$, then the …
  5. $\lim_{x\to0}\frac{\int_{0}^{x^{2}}\left(\sin\sqrt{t}\right)dt}{x^{3}}$ is equal to :
  6. $\lim_{n\to\infty}\left(1+\frac{1+\frac{1}{2}+\ldots+\frac{1}{n}}{n^{2}}\right)^{n}$ is equal to :
  7. …
  8. $\lim_{x\to\frac{\pi}{2}}\left(\frac{1}{\left(x-\frac{\pi}{2}\right)^{2}}\int_{x^{3}}^{\left(\frac{\pi}{2}\right)^{3}}\cos\left(t^{\frac{1}{3}}\right)dt\right)$ is equal …
  9. The value of the limit $\lim_{x\to4}\left(2^{x}+2^{5-x}-17\right)^{\frac{1}{8-2^{3x/4}}}$ is equal to
  10. Let $f:\mathbb{R}\to(0,\infty)$ be strictly increasing function such that $\lim_{x\to\infty}\frac{f(7x)}{f(x)}=1$. Then, the value of …
  11. $\lim_{x\to0}\left(\left(\frac{1-\cos^{2}(3x)}{\cos^{3}(4x)}\right)\left(\frac{\sin^{3}(4x)}{\left(\log_e(2x+1)\right)^{5}}\right)\right)$ is equal to
  12. If $\alpha>\beta>0$ are the roots of the equation $ax^{2}+bx+1=0$, and …
  13. If $\lim_{x\to0}\frac{\cos(2x)+a\cos(4x)-b}{x^{4}}$ is finite, then $(a+b)$ is equal to :
  14. Let $\alpha\in\mathbf{R}$ be such that the function …
  15. Let $f(x)=\begin{cases}x-1, & x\text{ is even},\\ 2x, & x\text{ is odd},\end{cases}$ $x\in\mathbf{N}$. If for some $a\in\mathbf{N}$, $f(f(f(a)))=21$, then …
  16. $\lim_{x\to0}\frac{e-(1+2x)^{\frac{1}{2x}}}{x}$ is equal to
  17. If $\lim_{x\to\infty}\left(\sqrt{x^{2}-x+1}-ax\right)=b$, then the ordered pair $(a,b)$ is :
  18. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as …
  19. …
  20. The value of the limit $\lim_{\theta\to0}\frac{\tan\left(\pi\cos^{2}\theta\right)}{\sin\left(2\pi\sin^{2}\theta\right)}$ is equal to :
  21. Suppose that the function $f(x)=\begin{cases}x^{2}-[x]+2a; & x<-1\\ 4x+5-b; & -1\le x<2\\ 3x^{2}+6[x]-2a; & x\ge2\end{cases}$ is continuous on $(-2,3)$ for some values …
  22. $\lim_{n\to\infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots+\frac{n}{(2n-1)^{2}}\right]$ is equal to :
  23. $\lim_{x\to0}\frac{\sin^{2}\left(\pi\cos^{4}x\right)}{x^{4}}$ is equal to :
  24. If $\alpha=\lim_{x\to\pi/4}\frac{\tan^{3}x-\tan x}{\cos\left(x+\frac{\pi}{4}\right)}$ and $\beta=\lim_{x\to0}(\cos x)^{\cot x}$ are the roots of the equation …
  25. For $\alpha,\beta,\gamma\in\mathbf{R}$, if $\lim_{x\to0}\frac{x^{2}\sin\alpha x+(\gamma-1)e^{x^{2}}}{\sin2x-\beta x}=3$, then $\beta+\gamma-\alpha$ is equal to :
  26. Among (S1) : $\lim_{n\to\infty}\frac{1}{n^{2}}(2+4+6+\ldots+2n)=1$ (S2) : $\lim_{n\to\infty}\frac{1}{n^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots+n^{15}\right)=\frac{1}{16}$
  27. If $\alpha,\beta$ are the distinct roots of $x^{2}+bx+c=0$, then $\lim_{x\to\beta}\frac{e^{2\left(x^{2}+bx+c\right)}-1-2\left(x^{2}+bx+c\right)}{(x-\beta)^{2}}$ is equal …
  28. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}\frac{\lambda|x^{2}-5x+6|}{\mu(5x-x^{2}-6)}, & x 2\\ \mu, & x=2\end{cases}$ where $[x]$ is the greatest …
  29. The value of $\lim_{h\to0}2\left\{\frac{\sqrt{3}\sin\left(\frac{\pi}{6}+h\right)-\cos\left(\frac{\pi}{6}+h\right)}{\sqrt{3}h\left(\sqrt{3}\cos h-\sin h\right)}\right\}$ …
  30. Let $f:\mathbf{R}\to\mathbf{R}$ be a function defined as $f(x)=\begin{cases}\frac{\sin(a+1)x+\sin2x}{2x}, & \text{if } x 0\end{cases}$ If $f$ is continuous at $x=0$ …
  31. For $a,b>0$, let $f(x)=\begin{cases}\frac{\tan((a+1)x)+b\tan x}{x}, & x 0\end{cases}$ be a continous function at $x=0$. Then $\frac{b}{a}$ is equal to :
  32. The value of $\lim_{n\to\infty}\frac{1}{n}\sum_{r=0}^{2n-1}\frac{n^{2}}{n^{2}+4r^{2}}$ is :
  33. Let $f:\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\to\mathbf{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If …
  34. Let $f(x)$ be a differentiable function at $x=a$ with $f'(a)=2$ and $f(a)=4$. Then $\lim_{x\to a}\frac{xf(a)-af(x)}{x-a}$ equals :
  35. Consider the function $f(x)=\begin{cases}\frac{a\left(7x-12-x^{2}\right)}{b\left|x^{2}-7x+12\right|}, & x 3\\ b, & x=3\end{cases}$ where $[x]$ denotes the greatest …
  36. $\lim_{x\to0}\frac{e^{2|\sin x|}-2|\sin x|-1}{x^{2}}$
  37. If for non-zero distinct real numbers $a,b$ and $c$ …
  38. The value of $\lim_{n\to\infty}\frac{[r]+[2r]+\ldots+[nr]}{n^{2}}$, where $r$ is a non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r$ …
  39. Let $f(x)=\left[x^{2}-x\right]+|-x+[x]|$, where $x\in\mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is
  40. If $U_n=\left(1+\frac{1}{n^{2}}\right)\left(1+\frac{2^{2}}{n^{2}}\right)^{2}\ldots\left(1+\frac{n^{2}}{n^{2}}\right)^{n}$, then …
  41. Let $a_1,a_2,a_3,\ldots,a_n$ be $n$ positive consecutive terms of an arithmetic progression. If $d>0$ is its common difference, then …
  42. $\lim_{x\to2}\left(\sum_{n=1}^{9}\frac{x}{n(n+1)x^{2}+2(2n+1)x+4}\right)$ is equal to :
  43. $\lim_{x\to2}\left(x^{3}-x^{2}-8x+13\right)^{\frac{1}{x^{3}-3x^{2}+4}}$
  44. The value of $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n^{3}}{\left(n^{2}+k^{2}\right)\left(n^{2}+3k^{2}\right)}$ is :
  45. Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f(x)=x-[x]$, $g(x)=1-x+[x]$, and $h(x)=\min\{f(x),g(x)\}$, $x\in[-2,2]$. Then $h$ is :
  46. Let $g(x)$ be a linear function and $f(x)=\begin{cases}g(x), & x\le0\\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}}, & x>0\end{cases}$, is continuous at $x=0$. If …
  47. If $a=\lim_{x\to0}\frac{\sqrt{1+\sqrt{1+x^{4}}}-\sqrt{2}}{x^{4}}$ and $b=\lim_{x\to0}\frac{\sin^{2}x}{\sqrt{2}-\sqrt{1+\cos x}}$, then the value of $ab^{3}$ is :
  48. The lowest integer which is greater than $\left(1+\frac{1}{10^{100}}\right)^{10^{100}}$ is :
  49. …
  50. The value of $\lim_{x\to0^{+}}\frac{\cos^{-1}\left(x-[x]^{2}\right)\cdot\sin^{-1}\left(x-[x]^{2}\right)}{x-x^{3}}$, where $[x]$ denotes the greatest integer $\le x$ is :

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