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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 151-200 of 282

  1. For each $x\in\mathbb R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\lim_{x\to0^-}\frac{x([x]+|x|)\sin[x]}{|x|}$ is equal to
  2. Let $x=2$ be a root of the equation $x^{2}+px+q=0$ and $f(x)=\begin{cases}\frac{1-\cos\left(x^{2}-4px+q^{2}+8q+16\right)}{(x-2p)^{4}},&x\neq2p\\0,&x=2p\end{cases}$. Then …
  3. Let $P=\lim_{x\to0^{+}}\left(1+\tan^{2}\sqrt x\right)^{\frac{1}{2x}}$, then $\log P$ is equal to
  4. Let $f:R\to R$ be a function defined as $f(x)=\begin{cases}5,&x\le1\\a+bx,&1<x<3\\b+5x,&3\le x<5\\30,&x\ge5\end{cases}$ Then $f$ is:
  5. The value of $\lim_{y\to0}\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt2}{y^4}$
  6. The value of $\lim_{x\to0}\frac1x\left[\tan^{-1}\left(\frac{x+1}{2x+1}\right)-\frac\pi4\right]$ is :
  7. $\lim_{x\to0}\frac{(1-\cos2x)(3+\cos x)}{x\tan4x}$ is equal to
  8. $\lim_{x\to0}\frac{(1-\cos2x)^{2}}{2x\tan x-x\tan2x}$ is
  9. Let $\alpha$ and $\beta$ be the distinct roots of $ax^{2}+bx+c=0$, then $\lim_{x\to\alpha}\frac{1-\cos\left(ax^{2}+bx+c\right)}{(x-\alpha)^{2}}$ is equal to
  10. $\lim_{x\to\infty}\left(\frac{x^{2}+5x+3}{x^{2}+x+2}\right)^{x}$
  11. The function $f:R/\{0\}\to R$ given by $f(x)=\frac1x-\frac{2}{e^{2x}-1}$ can be made continuous at $x=0$ by defining $f(0)$ as
  12. Let $f(2)=4$ and $f'(x)=4$. Then $\lim_{x\to2}\frac{xf(2)-2f(x)}{x-2}$ is given by
  13. If the function $f(x)=\begin{cases}\frac{\sqrt{2+\cos x}-1}{(\pi-x)^{2}}, & x\neq\pi\\ k, & x=\pi\end{cases}$ is continuous at $x=\pi$, then $k$ equals
  14. $\lim_{x\to0}\frac{\log x^{n}-[x]}{[x]}$, $n\in N$, ($[x]$ denotes the greatest integer less than or equal to $x$)
  15. Let $f(a)=g(a)=k$ and their $n^{\text{th}}$ derivatives $f^{n}(a)$, $g^{n}(a)$ exist and are not equal for some $n$. Further if …
  16. Let $f(x)=\frac{1-\tan x}{4x-\pi}$, $x\neq\frac\pi4$, $x\in\left[0,\frac\pi2\right]$. If $f(x)$ is continuous in $\left[0,\frac\pi2\right]$, then …
  17. $\lim_{x\to0}\frac{(1-\cos2x)(3+\cos x)}{x\tan4x}$ is equal to
  18. The value of $p$ and $q$ for which the function $f(x)=\begin{cases}\frac{\sin(p+1)x+\sin x}{x},&x 0\end{cases}$ is continuous for all $x$ in $R$, are
  19. If $\lim_{x\to2}\frac{\tan\left((x-2)\left(x^2+(k-2)x-2k\right)\right)}{x^2-4x+4}=5$, then $k$ is equal to :
  20. If $f(1)=1$, $f'(1)=2$, then $\lim_{x\to1}\frac{\sqrt{f(x)}-1}{\sqrt x-1}$ is
  21. The value of $\lim_{x\to0}\frac{\int_{0}^{x^{2}}\sec^{2}t\,dt}{x\sin x}$ is
  22. $\lim_{x\to0}\frac{\sqrt{1-\cos2x}}{\sqrt2\,x}$ is
  23. Let $f(x)=\begin{cases}(x-1)\sin\frac{1}{x-1}&\text{if }x\neq1\\0&\text{if }x=1\end{cases}$ Then which one of the following is true?
  24. $\lim_{x\to2}\left(\frac{\sqrt{1-\cos\{2(x-2)\}}}{x-2}\right)$
  25. $\lim_{x\to\frac\pi2}\frac{\left[1-\tan\left(\frac x2\right)\right][1-\sin x]}{\left[1+\tan\left(\frac x2\right)\right][\pi-2x]^{3}}$ is
  26. Let $f:R\to R$ be a differentiable function having $f(2)=6$, $f'(2)=\frac{1}{48}$. Then $\lim_{x\to2}\int_{6}^{f(x)}\frac{4t^{3}}{x-2}\,dt$ equals :
  27. Consider the function : $f(x)=[x]+|1-x|$, $-1\le x\le3$ where $[x]$ is the greatest integer function. Statement 1: $f$ is not continuous at $x=0,1,2$ and $3$. Statement …
  28. If the function $g(x)=\begin{cases}k\sqrt{x+1},&0\le x\le3\\mx+2,&3<x\le5\end{cases}$ is differentiable, then the value of $k+m$ is :
  29. If $\lim_{x\to\infty}\left(1+\frac ax+\frac{b}{x^{2}}\right)^{2x}=e^{2}$, then the value of $a$ and $b$, are
  30. Let $a,b\in R,\ (a\neq0)$. If the function $f$, defined as …
  31. If $f(x)=\begin{cases}xe^{-\left(\frac{1}{|x|}+\frac1x\right)},&x\neq0\\0,&x=0\end{cases}$ then $f(x)$ is
  32. If $\lim_{x\to0}\frac{\log(3+x)-\log(3-x)}{x}=k$, the value of $k$ is
  33. If $f(x)$ is continuous and $f\left(\frac92\right)=\frac29$, then $\lim_{x\to0}f\left(\frac{1-\cos3x}{x^{2}}\right)$ equals to
  34. Let $f:(0,1)\to\mathbb{R}$ be the function defined as $f(x)=[4x]\left(x-\frac14\right)^2\left(x-\frac12\right)$, where $[x]$ denotes the greatest integer less than or …
  35. Let $\alpha$ be a positive real number. Let $f:\mathbb{R}\to\mathbb{R}$ and $g:(\alpha,\infty)\to\mathbb{R}$ be the functions defined by $$f(x)=\sin\left(\frac{\pi …
  36. $\lim_{n\to\infty}\left(\frac1{1+n}+\frac1{2+n}+\frac1{3+n}+\ldots+\frac1{2n}\right)$ is equal to :
  37. $\lim_{n\to\infty}\left(\frac{(n+1)^{1/3}}{n^{4/3}}+\frac{(n+2)^{1/3}}{n^{4/3}}+\ldots+\frac{(2n)^{1/3}}{n^{4/3}}\right)$ is equal to
  38. Let $f:(0,1)\to\mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x\in\left[\frac{1}{n+1},\frac1n\right)$ where $n\in\mathbb{N}$. Let $g:(0,1)\to\mathbb{R}$ be a …
  39. If $\lim_{x\to\infty}\left(1+\frac ax-\frac{4}{x^{2}}\right)^{2x}=e^{3}$, then $a$ is equal to
  40. If $f(x)=\begin{cases}\frac{\sin(p+1)x+\sin x}{x}, & x 0\end{cases}$ is continuous at $x=0$, then the ordered pair $(p,q)$ is equal to :
  41. Let $f(x)=5-|x-2|$ and $g(x)=|x+1|$, $x\in\mathbb R$. If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta$, then …
  42. For $a\in\mathbb{R}$, $|a|>1$, let …
  43. Let $f:R-\{0\}\to R$ be a function such that $f(x)-6f\left(\frac1x\right)=\frac{35}{3x}-\frac52$. If the $\lim_{x\to0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta$ …
  44. If $\lim_{x\to1}\frac{x^{4}-1}{x-1}=\lim_{x\to k}\frac{x^{3}-k^{3}}{x^{2}-k^{2}}$, then $k$ is
  45. If $$\beta=\lim_{x\to 0}\frac{e^{x^{3}}-(1-x^{3})^{\frac{1}{3}}+\left((1-x^{2})^{\frac{1}{2}}-1\right)\sin x}{x\sin^{2}x},$$ then the value of $6\beta$ is ________.
  46. …
  47. The set of values of $a$ for which $\lim_{x\to a}\left([x-5]-[2x+2]\right)=0$, where $[\zeta]$ denotes the greatest integer less than or equal to $\zeta$, is equal to
  48. $\lim_{n\to\infty}\left(\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\frac{n}{n^{2}+3^{2}}+\ldots+\frac{1}{5n}\right)$ is equal to
  49. Let $S$ be the set of all $(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}$ such that $$\lim_{x\to\infty}\frac{\sin(x^2)(\log_e …
  50. $\lim_{x\to0}\frac{x+2\sin x}{\sqrt{x^{2}+2\sin x+1}-\sqrt{\sin^{2}x-x+1}}$ is

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