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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 101-150 of 282

  1. $\lim_{x\to0}\left(\frac{3x^2+2}{7x^2+2}\right)^{1/x^2}$ is equal to :
  2. Let $\beta=\lim_{x\to0}\frac{\alpha x-(e^{3x}-1)}{\alpha x(e^{3x}-1)}$ for some $\alpha\in\mathbb{R}$. Then the value of $\alpha+\beta$ is :
  3. Let $f(x)=\begin{cases}x^{3}+8; & x<0\\ x^{2}-4; & x\ge0\end{cases}$ and $g(x)=\begin{cases}(x-8)^{1/3}; & x<0\\ (x+4)^{1/2}; & x\ge0\end{cases}$. Then the number of …
  4. $\lim_{x\to\frac{1}{\sqrt{2}}}\frac{\sin\left(\cos^{-1}x\right)-x}{1-\tan\left(\cos^{-1}x\right)}$ is equal to :
  5. …
  6. If $\lim_{x\to0}\frac{ax^{2}e^{x}-b\log_e(1+x)+cxe^{-x}}{x^{2}\sin x}=1$, then $16\left(a^{2}+b^{2}+c^{2}\right)$ is equal to $\_\_\_\_\_\_$ .
  7. For $t>-1$, let $\alpha_t$ and $\beta_t$ be the roots of the equation $\left((t+2)^{1/7}-1\right)x^{2}+\left((t+2)^{1/6}-1\right)x+\left((t+2)^{1/21}-1\right)=0$. If …
  8. If $\lim_{x\to1}\frac{\sin\left(3x^{2}-4x+1\right)-x^{2}+1}{2x^{3}-7x^{2}+ax+b}=-2$, then the value of $(a-b)$ is equal to $\_\_\_\_\_\_$ .
  9. Let …
  10. 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 …
  11. If $\lim_{x\to0}\left(\frac{\tan x}{x}\right)^{\frac{1}{x^{2}}}=p$, then $96\log_ep$ is equal to $\_\_\_\_\_\_$ .
  12. If $\lim_{x\to1}\frac{x+x^2+x^3+\ldots+x^n-n}{x-1}=820$, ($n\in\mathbf{N}$) then the value of $n$ is equal to ________.
  13. Let $[t]$ denote the greatest integer less than or equal to $t$. Let $f:[0,\infty)\to\mathbf{R}$ be a function defined by $f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$. …
  14. Let $f(x)=\left[2x^{2}+1\right]$ and $g(x)=\begin{cases}2x-3, & x<0\\ 2x+3, & x\ge0\end{cases}$, where $[t]$ is the greatest integer $\le t$. Then, in the open interval …
  15. If $\alpha=\lim_{x\to0^{+}}\left(\frac{e^{\sqrt{\tan x}}-e^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta=\lim_{x\to0}(1+\sin x)^{\frac{1}{2}\cot x}$ are the …
  16. If $\lim_{x\to0}\left\{\frac1{x^8}\left(1-\cos\frac{x^2}2-\cos\frac{x^2}4+\cos\frac{x^2}2\cos\frac{x^2}4\right)\right\}=2^{-k}$, then the value of $k$ is ________.
  17. If the function $f(x)=\frac{\tan(\tan x)-\sin(\sin x)}{\tan x-\sin x}$ is continuous at $x=0$, then $f(0)$ is equal to $\_\_\_\_\_\_$ .
  18. Let $\{x\}$ denote the fractional part of $x$ and $f(x)=\frac{\cos^{-1}\left(1-\{x\}^{2}\right)\sin^{-1}(1-\{x\})}{\{x\}-\{x\}^{3}}$, $x\neq0$. If $L$ and $R$ …
  19. If $\lim_{n\to\infty}\frac{(n+1)^{k-1}}{n^{k+1}}\left[(nk+1)+(nk+2)+\ldots+(nk+n)\right]=33\cdot\lim_{n\to\infty}\frac{1}{n^{k+1}}\left[1^k+2^k+3^k+\ldots+n^k\right]$ …
  20. Suppose $\lim_{x\to0}\frac{F(x)}{x^{3}}$ exists and is equal to $L$, where …
  21. Let $f:(0,\pi)\to\mathbf{R}$ be a function given by …
  22. The number of points where the function …
  23. $\lim_{x\to0}\left(\frac{(x+2\cos x)^3+2(x+2\cos x)^2+3\sin(x+2\cos x)}{(x+2)^3+2(x+2)^2+3\sin(x+2)}\right)^{\frac{100}{x}}$ is equal to ________.
  24. The number of points of discontinuity of the function $f(x)=\left[\frac{x^{2}}{2}\right]-\left[\sqrt{x}\right]$, $x\in[0,4]$, where $[\cdot]$ denotes the greatest …
  25. The value of $\lim_{x\to0}2\left(\frac{1-\cos x\sqrt{\cos2x}\sqrt[3]{\cos3x}\ldots\ldots\sqrt[10]{\cos10x}}{x^{2}}\right)$ is $\_\_\_\_\_\_$ .
  26. The value of $\lim_{n\to\infty}\frac{1+2-3+4+5-6+\ldots+(3n-2)+(3n-1)-3n}{\sqrt{2n^{4}+4n+3}-\sqrt{n^{4}+5n+4}}$ is
  27. If $f:R\to R$ is a differentiable function and $f(2)=6$, then $\lim_{x\to2}\int_{6}^{f(x)}\frac{2t\,dt}{x-2}$ is :
  28. If the function $f(x)=\begin{cases}(1+|\cos x|)^{\frac{\lambda}{|\cos x|}},&0<x<\frac\pi2\\\mu,&x=\frac\pi2\\e^{\frac{\cot6x}{\cot4x}},&\frac\pi2<x<\pi\end{cases}$ is …
  29. For each $t\in R$, let $[t]$ be the greatest integer less than or equal to $t$. Then …
  30. $\lim_{x\to0}\frac{\sin^{2}x}{\sqrt2-\sqrt{1+\cos x}}$ equals
  31. The value of $k$ which the function $f(x)=\begin{cases}\left(\frac45\right)^{\frac{\tan4x}{\tan5x}}, & 0<x<\frac\pi2\\ k+\frac25, & x=\frac\pi2\end{cases}$ is continuous …
  32. $\lim_{x\to\frac{\pi}{2}}\frac{\cot x-\cos x}{(\pi-2x)^3}$ equals :
  33. $\lim_{x\to0}\frac{e^{x^2}-\cos x}{\sin^2x}$ is equal to
  34. If $\lim_{x\to0}\frac{3+\alpha\sin x+\beta\cos x+\log_e(1-x)}{3\tan^2x}=\frac13$, then $2\alpha-\beta$ is equal to :
  35. Let $f(x)$ be a differentiable function such that $f'(x)=7-\frac34\frac{f(x)}{x}$, $(x>0)$ and $f(1)\neq4$. Then $\lim_{x\to0^+}x\,f\left(\frac1x\right)$ :
  36. $\lim_{x\to3}\frac{\sqrt{3x}-3}{\sqrt{2x-4}-\sqrt2}$ is equal to :
  37. $\lim_{x\to0}\frac{x\cot(4x)}{\sin^{2}x\cot^{2}(2x)}$ is equal to :
  38. Let $f,g:R\to R$ be two functions defined by $f(x)=\begin{cases}x\sin\left(\frac1x\right),&x\neq0\\0,&x=0\end{cases}$ and $g(x)=xf(x)$. Statement I: $f$ is a continuous …
  39. If $\lim_{n\to\infty}\left(\frac{1^{a}+2^{a}+\ldots+n^{a}}{(n+1)^{a-1}\left[(na+1)+(na+2)+\ldots+(na+n)\right]}\right)=\frac1{60}$ for some positive real number $a$ …
  40. Let $[x]$ denote the greatest integer less than or equal to $x$. Then …
  41. If the function $f$ defined on $\left(\frac\pi6,\frac\pi3\right)$ by $f(x)=\begin{cases}\frac{\sqrt2\cos x-1}{\cot x-1},&x\neq\frac\pi4\\k,&x=\frac\pi4\end{cases}$ is …
  42. Let $f:\mathbb R\to\mathbb R$ be a differentiable function satisfying $f'(3)+f'(2)=0$. Then $\lim_{x\to0}\left(\frac{1+f(3+x)-f(3)}{1+f(2-x)-f(2)}\right)^{\frac1x}$ is …
  43. If $f(x)=[x]-\left[\frac x4\right]$, $x\in R$, where $[x]$ denotes the greatest integer function, then :
  44. If the function $f(x)=\begin{cases}a|\pi-x|+1, & x\le5\\ b|x-\pi|+3, & x>5\end{cases}$ is continuous at $x=5$, then the value of $a-b$ is :
  45. Let a function $f:R\to R$ be defined as, $f(x)=\begin{cases}\sin x-e^{x} & \text{if } x\le0\\ a+[-x] & \text{if } 0<x<1\\ 2x-b & \text{if } x\ge1\end{cases}$ where $[x]$ …
  46. $\lim_{x\to0}\frac{x\tan2x-2x\tan x}{(1-\cos2x)^{2}}$ equals.
  47. $\lim_{n\to\infty}\left(\frac{(n+1)(n+2)\ldots3n}{n^{2n}}\right)^{\frac1n}$ is equal to
  48. For each $t\in R$, let $[t]$ be the greatest integer less than or equal to $t$. Then …
  49. Let $k$ be a non-zero real number. If $f(x)=\begin{cases}\frac{(e^{x}-1)^{2}}{\sin\left(\frac xk\right)\log\left(1+\frac x4\right)}, & x\neq0\\ 12, & x=0\end{cases}$ is …
  50. Let $f(x)=\begin{cases}(x-1)^{\frac{1}{2-x}}, & x>1,\ x\neq2\\ k, & x=2\end{cases}$ The value of $k$ for which $f$ is continuous at $x=2$ is

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