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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 51-100 of 282

  1. $\lim_{x\to0}(1+x)^{\frac{1}{e^{x}-1}}$ is equal to
  2. If $\lim_{x\to0}\frac{e^{ax}-\cos(bx)-\frac{cxe^{-cx}}{2}}{1-\cos(2x)}=17$, then $5a^{2}+b^{2}$ is equal to
  3. If $\lim_{x\to0}\frac{\sin^{-1}x-\tan^{-1}x}{3x^{3}}$ is equal to $L$, then the value of $(6L+1)$ is :
  4. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}2\sin\left(-\frac{\pi x}{2}\right), & \text{if } x 1\end{cases}$ If $f(x)$ is continuous on …
  5. If the function $f(x)=\begin{cases}\frac{1}{x}\log_e\left(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}\right), & x 0\end{cases}$ is continuous at $x=0$, then …
  6. If $f(x)$ be a differentiable function such that $f(9)=9$ and $f'(9)=4$, then $\lim_{x\to9}\frac{\sqrt{f(x)}-3}{\sqrt{x}-3}$ is $\_\_\_\_\_\_$ .
  7. Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined as $f(x)=a\sin\left(\frac{\pi[x]}{2}\right)+[2-x]$, $a\in\mathbb{R}$, where $[t]$ is the greatest integer less than …
  8. Let $f:\mathbf{R}\to\mathbf{R}$ be a continuous function such that $f(3x)-f(x)=x$. If $f(8)=7$, then $f(14)$ is equal to :
  9. If $\lim_{x\to0}\frac{ax-\left(e^{4x}-1\right)}{ax\left(e^{4x}-1\right)}$ exists and is equal to $b$, then the value of $a-2b$ is $\_\_\_\_\_\_$ .
  10. The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\lim_{n\to\infty}\frac{\cos(2\pi x)-x^{2n}\sin(x-1)}{1+x^{2n+1}-x^{2n}}$ is continuous for all $x$ in :
  11. Let $f:\left(-\frac\pi4,\frac\pi4\right)\to\mathbf{R}$ be defined as …
  12. Let $f,g:\mathbf{R}\to\mathbf{R}$ be functions defined by $f(x)=\begin{cases}[x], & x<0\\ |1-x|, & x\ge0\end{cases}$ and …
  13. Let $f:\mathbf{R}\to\mathbf{R}$ be a function such that $f(2)=4$ and $f'(2)=1$. Then, the value of $\lim_{x\to2}\frac{x^2f(2)-4f(x)}{x-2}$ is equal to :
  14. Let $f(x)=\frac{x}{\left(1+x^{n}\right)^{\frac{1}{n}}}$, $x\in\mathbb{R}-\{-1\}$, $n\in\mathbb{N}$, $n>2$. If …
  15. If for $p\ne q\ne0$, the function $f(x)=\frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then :
  16. Let $a,b\in\mathbf{R}$, $b\neq0$. Define a function …
  17. $\lim_{x\to0}\frac{\int_0^xt\sin(10t)\,dt}{x}$ is equal to :
  18. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as …
  19. Let $f(x)$ be a polynomial function such that $f(x)+f'(x)+f''(x)=x^{5}+64$. Then, the value of $\lim_{x\to1}\frac{f(x)}{x-1}$ is equal to :
  20. $\lim_{x\to0}\left(\tan\left(\frac\pi4+x\right)\right)^{1/x}$ is equal to :
  21. If $\lim_{n\to\infty}\left(\sqrt{n^2-n-1}+n\alpha+\beta\right)=0$, then $8(\alpha+\beta)$ is equal to :
  22. Let $f(x)=\begin{cases}(1+ax)^{1/x}, & x 0\end{cases}$ be continuous at $x=0$. Then $e^{a}bc$ is equal to:
  23. If $\alpha$ is the positive root of the equation, $p(x)=x^2-x-2=0$, then $\lim_{x\to\alpha^+}\frac{\sqrt{1-\cos(p(x))}}{x+\alpha-4}$ is equal to :
  24. Let $f:\mathbf{R}\to\mathbf{R}$ be a differentiable function such that $f\left(\frac{\pi}{4}\right)=\sqrt{2}$, $f\left(\frac{\pi}{2}\right)=0$ and …
  25. The number of points in the interval $[2,4]$, at which the function $f(x)=\left[x^{2}-x-\frac{1}{2}\right]$, where $[\cdot]$ denotes the greatest integer function, is …
  26. $\lim_{x\to0}\frac{\cos(\sin x)-\cos x}{x^{4}}$ is equal to :
  27. If $f(x)=\begin{cases}x+a, & x\le0\\ |x-4|, & x>0\end{cases}$ and $g(x)=\begin{cases}x+1, & x<0\\ (x-4)^2+b, & x\ge0\end{cases}$ are continuous on $\mathbf{R}$, then …
  28. If $\lim_{x\to1^{+}}\frac{(x-1)(6+\lambda\cos(x-1))+\mu\sin(1-x)}{(x-1)^{3}}=-1$, where $\lambda,\mu\in\mathbb{R}$, then $\lambda+\mu$ is equal to
  29. $\lim_{n\to\infty}\sum_{r=1}^{n}\frac{r}{2r^{2}-7rn+6n^{2}}$ is equal to :
  30. Consider the function $f(x)=\begin{cases}\frac{P(x)}{\sin(x-2)}, & x\neq2\\ 7, & x=2\end{cases}$ where $P(x)$ is a polynomial such that $P''(x)$ is always a constant and …
  31. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2x)-f(x)=x$ for all $x\in\mathbb{R}$. If …
  32. The value of $\lim_{x\to0}\left(\frac{x}{\sqrt[8]{1-\sin x}-\sqrt[8]{1+\sin x}}\right)$ is equal to :
  33. Let $[t]$ denote the greatest integer $\le t$. If for some $\lambda\in\mathbf{R}-\{0,1\}$, $\lim_{x\to0}\left|\frac{1-x+|x|}{\lambda-x+[x]}\right|=L$, then $L$ is equal …
  34. Let the function $f(x)=\begin{cases}\dfrac{\log_e(1+5x)-\log_e(1+\alpha x)}{x}, & x\ne0\\ 10, & x=0\end{cases}$ be continuous at $x=0$. Then $\alpha$ is equal to
  35. If $f(x)=\begin{cases}\dfrac{\sin(a+2)x+\sin x}{x}, & x 0\end{cases}$ is continuous at $x=0$, then $a+2b$ is equal to :
  36. …
  37. Let $a$ be an integer such that $\lim_{x\to7}\frac{18-[1-x]}{[x-3a]}$ exists, where $[t]$ is greatest integer $\le t$. Then $a$ is equal to :
  38. Let $f$ be a differentiable function on $\mathbb{R}$ such that $f(2)=1$, $f'(2)=4$. Let $\lim_{x\to0}\left(f(2+x)\right)^{3/x}=e^{\alpha}$. Then the number of times the …
  39. Given below are two statements: Statement I : $\lim_{x\to0}\left(\frac{\tan^{-1}x+\log_e\sqrt{\frac{1+x}{1-x}}-2x}{x^{5}}\right)=\frac{2}{5}$ Statement II …
  40. If the function $f(x)=\begin{cases}\dfrac{\log_e(1-x+x^2)+\log_e(1+x+x^2)}{\sec x-\cos x}, & x\in\left(-\frac\pi2,\frac\pi2\right)-\{0\}\\ k, & x=0\end{cases}$ is …
  41. If $\lim_{x\to0}\frac{\alpha e^x+\beta e^{-x}+\gamma\sin x}{x\sin^2x}=\frac23$, where $\alpha,\beta,\gamma\in\mathbf{R}$, then which of the following is NOT correct ?
  42. $\lim_{x\to\frac\pi4}\frac{8\sqrt2-(\cos x+\sin x)^7}{\sqrt2-\sqrt2\sin2x}$ is equal to
  43. $\lim_{x\to a}\frac{(a+2x)^{\frac13}-(3x)^{\frac13}}{(3a+x)^{\frac13}-(4x)^{\frac13}}$ ($a\ne0$) is equal to :
  44. Let $f(x)=\begin{cases}\dfrac{\sin(ax+3x)+\sin x}{x}, & x 0\end{cases}$ be continuous at $x=0$. If $(1-k)f\left(\frac7{18}\right)=ab$, then $k$ is equal to
  45. $\lim_{x\to\frac{\pi}{2}}\left(\tan^{2}x\left(\left(2\sin^{2}x+3\sin x+4\right)^{\frac{1}{2}}-\left(\sin^{2}x+6\sin x+2\right)^{\frac{1}{2}}\right)\right)$ is equal to
  46. $\lim_{x\to0^{+}}\frac{\tan\left(5(x)^{\frac{1}{3}}\right)\log_e\left(1+3x^{2}\right)}{\left(\tan^{-1}3\sqrt{x}\right)^{2}\left(e^{5(x)^{\frac{4}{3}}}-1\right)}$ is …
  47. Let $[x]$ be the greatest integer $\le x$. Then the number of points in the interval $(-2,1)$, where the function $f(x)=\big|[x]\big|+\sqrt{x-[x]}$ is discontinuous, is …
  48. If the function $f(x)=\frac{\cos(\sin x)-\cos x}{x^{4}}$ is continuous at each point in its domain and $f(0)=\frac{1}{k}$, then $k$ is $\_\_\_\_\_\_$ .
  49. If a function $f(x)$ defined by $f(x)=\begin{cases}ae^x+be^{-x}, & -1\le x<1\\ cx^2, & 1\le x\le3\\ ax^2+2cx, & 3<x\le4\end{cases}$ be continuous for some …
  50. The value of $\lim_{x\to1}\frac{\left(x^{2}-1\right)\sin^{2}(\pi x)}{x^{4}-2x^{3}+2x-1}$ is equal to:

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