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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 201-250 of 282

  1. $\lim_{t\to0}\left(1^{\frac{1}{\sin^{2}t}}+2^{\frac{1}{\sin^{2}t}}+3^{\frac{1}{\sin^{2}t}}+\ldots+n^{\frac{1}{\sin^{2}t}}\right)^{\sin^{2}t}$ is equal to
  2. $\lim_{x\to1^-}\frac{\sqrt\pi-\sqrt{2\sin^{-1}x}}{\sqrt{1-x}}$ is equal to
  3. $\lim_{x\to\frac\pi4}\frac{\cot^{3}x-\tan x}{\cos\left(x+\frac\pi4\right)}$ is
  4. Let $\alpha$ and $\beta$ be the real numbers such that $$\lim_{x\to0}\frac1{x^3}\left(\frac{\alpha}{2}\int_0^x\frac1{1-t^2}\,dt+\beta x\cos x\right)=2.$$ Then the value …
  5. Let $x_0$ be the real number such that $e^{x_0}+x_0=0$. For a given real number $\alpha$, define $$g(x)=\frac{3xe^x+3x-\alpha e^x-\alpha x}{3(e^x+1)}$$ for all real …
  6. If $\lim_{x\to1}\frac{x^{2}-ax+b}{x-1}=5$, then $a+b$ is equal to :
  7. $\lim_{n\to\infty}\frac3n\left\{4+\left(2+\frac1n\right)^{2}+\left(2+\frac2n\right)^{2}+\ldots+\left(3-\frac1n\right)^{2}\right\}$ is equal to
  8. Let $f:\mathbb{R}\to\mathbb{R}$ be a function. We say that $f$ has PROPERTY 1 if $\lim_{h\to0}\frac{f(h)-f(0)}{\sqrt{|h|}}$ exists and is finite, and PROPERTY 2 if …
  9. Let $k\in\mathbb{R}$. If $\lim_{x\to0^+}\left(\sin(\sin kx)+\cos x+x\right)^{\frac{2}{x}}=e^6$, then the value of $k$ is
  10. $\lim_{x\to0}\frac{(27+x)^{\frac13}-3}{9-(27+x)^{\frac23}}$ equals
  11. If the function $f$ defined as $f(x)=\frac1x-\frac{k-1}{e^{2x}-1}$, $x\neq0$ is continuous at $x=0$, then the ordered pair $(k,f(0))$ is equal to
  12. Let $f,g$ and $h$ be the real valued functions defined on $\mathbb R$ as $f(x)=\begin{cases}\frac{x}{|x|},&x\neq0\\1,&x=0\end{cases}$ …
  13. $\lim_{x\to0}\operatorname{cosec}x\left(\sqrt{2\cos^{2}x+3\cos x}-\sqrt{\cos^{2}x+\sin x+4}\right)$ is :
  14. If the function $f$ defined on $\left(-\frac13,\frac13\right)$ by $f(x)=\begin{cases}\frac1x\log_e\left(\frac{1+3x}{1-2x}\right),&x\neq0\\k,&x=0\end{cases}$ is …
  15. Let $f(x)=x^{6}+2x^{4}+x^{3}+2x+3$, $x\in R$. Then the natural number $n$ for which $\lim_{x\to1}\frac{x^{n}f(1)-f(x)}{x-1}=44$ is ______.
  16. For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $S$, let $|S|$ denote the number of elements in …
  17. If $\lim_{t\to0}\left(\int_0^{1}(3x+5)^{t}dx\right)^{\frac1t}=\frac{\alpha}{5e}\left(\frac85\right)^{\frac23}$, then $\alpha$ is equal to
  18. Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p\in\mathbf N$ for which …
  19. Consider the function $f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to(-\infty,\infty)$ defined by $$f(x)=(|x|+|x-1|)\sin x+[x\sin x],$$ where $[x\sin x]$ is the greatest …
  20. If the value of $\lim_{x\to0}\left(2-\cos x\sqrt{\cos2x}\right)^{\left(\frac{x+2}{x^{2}}\right)}$ is equal to $e^{a}$, then $a$ is equal to
  21. $\lim_{x\to0}\frac{48}{x^4}\int_0^x\frac{t^3}{t^6+1}dt$ is equal to ______.
  22. $\lim_{x\to2}\frac{3^{x}+3^{3-x}-12}{3^{-x/2}-3^{1-x}}$ is equal to
  23. $f$ is defined in $[-5,5]$ as $f(x)=x$ if $x$ is rational, $f(x)=-x$ if $x$ is irrational. Then
  24. Let $f:R\to R$ be a positive increasing function with $\lim_{x\to\infty}\frac{f(3x)}{f(x)}=1$. Then $\lim_{x\to\infty}\frac{f(2x)}{f(x)}=$
  25. For $x>0$, $\displaystyle\lim_{x\to0}\left((\sin x)^{1/x}+(1/x)^{\sin x}\right)$ is
  26. If $f(x)=\min\{1,x^2,x^3\}$, then
  27. If $f(x)=\begin{cases}\dfrac{a|x|+x^2-2(\sin|x|)(\cos|x|)}{x}, & x\ne0\\ b, & x=0\end{cases}$ is continuous at $x=0$, then $a+b$ is equal to :
  28. Let $[\cdot]$ denote the greatest integer function and $f(x)=\lim_{n\to\infty}\frac{1}{n^3}\sum_{k=1}^{n}\left[\frac{k^2}{3^x}\right]$. Then $12\sum_{j=1}^{\infty}f(j)$ …
  29. The value of $\lim_{x\to0}\frac{\log_e\left(\sec(ex)\cdot\sec\left(e^2x\right)\cdot\ldots\cdot\sec\left(e^{10}x\right)\right)}{e^2-e^{2\cos x}}$ is equal to :
  30. 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 :
  31. The product of all possible values of $\alpha$, for which $\lim_{x\to0}\left(\frac{1-\cos(\alpha x)\cos((\alpha+1)x)\cos((\alpha+2)x)}{\sin^{2}((\alpha+1)x)}\right)=2$ …
  32. The value of $\lim_{x\to0}\left(\frac{x^{2}\sin^{2}x}{x^{2}-\sin^{2}x}\right)$ is :
  33. If $\lim_{x\to2}\frac{\sin\left(x^{3}-5x^{2}+ax+b\right)}{\left(\sqrt{x-1}-1\right)\log_{e}(x-1)}=m$, then $a+b+m$ is equal to :
  34. If $\lim_{x\to0}\frac{e^{(a-1)x}+2\cos(bx)+(c-2)e^{-x}}{x\cos x-\log_e(1+x)}=2$, then $a^2+b^2+c^2$ is equal to :
  35. Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min\{\sqrt2x,x^2\}$. Let …
  36. Let $\alpha,\beta\in\mathbf{R}$ be such that the function $f(x)=\begin{cases}2\alpha\left(x^2-2\right)+2\beta x, & x<1\\ (\alpha+3)x+(\alpha-\beta), & x\ge1\end{cases}$ …
  37. Let $[t]$ denote the greatest integer less than or equal to $t$. If the function …
  38. Let $f(x)$ be differentiable on the interval $(0,\infty)$ such that $f(1)=1$, and $$\lim_{t\to x}\frac{t^2 f(x) - x^2 f(t)}{t-x} = 1$$ for each $x>0$. Then $f(x)$ is
  39. $$\lim_{x\to\frac{\pi}{4}}\frac{\int_{2}^{\sec^2 x} f(t)\,dt}{x^2-\frac{\pi^2}{16}}$$ equals
  40. Let $f$ and $g$ be real valued functions defined on interval $(-1,1)$ such that $g''(x)$ is continuous, $g(0) eq0$, $g'(0)=0$, $g''(0) eq0$, and $f(x)=g(x)\sin x$. …
  41. Let $g(x)=\dfrac{(x-1)^n}{\log\cos^m(x-1)}$; $0 0$, and let $p$ be the left hand derivative of $|x-1|$ at $x=1$. If $\lim_{x\to1^+}g(x)=p$, then
  42. The largest value of the nonnegative integer $a$ for which $$\lim_{x\to1}\left\{\frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1}\right\}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$$ is
  43. Let $\alpha,\beta\in\mathbb{R}$ be such that $\lim_{x\to0}\frac{x^2\sin(\beta x)}{\alpha x-\sin x}=1$. Then $6(\alpha+\beta)$ equals
  44. Let $$L=\lim_{x\to0}\frac{a-\sqrt{a^2-x^2}-\frac{x^2}{4}}{x^4},\quad a>0.$$ If $L$ is finite, then
  45. The value of $\lim_{x\to0}\frac{1}{x^3}\int_0^x\frac{t\ln(1+t)}{t^4+4}\,dt$ is
  46. If $$f(x)=\begin{cases}-x-\frac{\pi}{2}, & x\le -\frac{\pi}{2}\\ -\cos x, & -\frac{\pi}{2} 1,\end{cases}$$ then
  47. If $$\lim_{x\to 0}\left[1+x\ln(1+b^2)\right]^{\frac{1}{x}}=2b\sin^2\theta,\ b>0\text{ and }\theta\in(-\pi,\pi],$$ then the value of $\theta$ is
  48. If $\displaystyle\lim_{x\to\infty}\left(\frac{x^2+x+1}{x+1}-ax-b\right)=4$, then
  49. Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation $(\sqrt[3]{1+a} - 1)x^2 + (\sqrt{1+a} - 1)x + (\sqrt[6]{1+a} - 1) = 0$ where $a > -1$. Then …
  50. For every integer $n$, let $a_n$ and $b_n$ be real numbers. Let function $f: \mathbb{R} \to \mathbb{R}$ be given by $$f(x) = \begin{cases} a_n + \sin\pi x, & \text{for } …

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