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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 251-282 of 282

  1. Let $m$ and $n$ be two positive integers greater than 1. If $$\lim_{\alpha\to0}\left(\frac{e^{\cos(\alpha^n)}-e}{\alpha^m}\right)=-\left(\frac{e}{2}\right)$$ then the …
  2. For each positive integer $n$, let $$y_n=\frac{1}{n}\left((n+1)(n+2)\cdots(n+n)\right)^{\frac{1}{n}}.$$ For $x\in\mathbb{R}$, let $[x]$ be the greatest integer less than …
  3. Let $f_1:\mathbb{R}\to\mathbb{R}$, $f_2:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to\mathbb{R}$, $f_3:\left(-1,e^{\pi/2}-2\right)\to\mathbb{R}$ and …
  4. Let $e$ denote the base of the natural logarithm. The value of the real number $a$ for which the right hand limit $$\lim_{x\to0^+}\frac{(1-x)^{\frac1x}-e^{-1}}{x^a}$$ is …
  5. The value of the limit $$\lim_{x\to\pi/2}\frac{4\sqrt2(\sin3x+\sin …
  6. Let the functions $f:(-1,1)\to\mathbb{R}$ and $g:(-1,1)\to(-1,1)$ be defined by $$f(x)=|2x-1|+|2x+1| \quad\text{and}\quad g(x)=x-[x],$$ where $[x]$ denotes the greatest …
  7. Let $f(x)=\begin{cases}3x, & x 2\end{cases}$, where $[\,.\,]$ denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where $f$ is not …
  8. $\lim_{x\to\infty}\frac{(2x^2-3x+5)(3x-1)^{\frac{x}{2}}}{(3x^2+5x+4)\sqrt{(3x+2)^x}}$ is equal to :
  9. $\lim_{x\to0}\frac{\sin\left(\pi\cos^{2}x\right)}{x^{2}}$ is equal to :
  10. If $f:\mathbf{R}\to\mathbf{R}$ is a function defined by $f(x)=[x]\cos\left(\frac{2x-1}{2}\right)\pi$, where $[x]$ denotes the greatest integer function, then $f$ is :
  11. Consider the function, $f(x)=|x-2|+|x-5|$, $x\in\mathbf{R}$. Statement 1 : $f'(4)=0$ Statement 2 : $f$ is continuous in $[2,5]$, differentiable in $(2,5)$ and …
  12. Let $\vec A(t)=f_1(t)\hat i+f_2(t)\hat j$ and $\vec B(t)=g_1(t)\hat i+g_2(t)\hat j$, $t\in[0,1]$, where $f_1,f_2,g_1,g_2$ are continuous functions. If $\vec A(t)$ and …
  13. The value of $\displaystyle\lim_{x\to2}\left(2-\cos\left(2x^{2}-5x+2\right)\right)^{\frac{1}{\left(x^{2}-4x+4\right)}}$ is equal to
  14. Let the functions $f:\mathbf{R}\to\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be defined as : $f(x)=\begin{cases}x+2, & x<0\\ x^{2}, & x\ge0\end{cases}$ and …
  15. If $\displaystyle\lim_{x\to0}\dfrac{ae^{x}-b\cos x+ce^{-x}}{x\sin x}=2$, then $a+b+c$ is equal to $\_\_\_\_\_\_\_\_$ .
  16. Let $f:\mathbf{R}\to\mathbf{R}$ be a function given by $f(x)=\begin{cases}\dfrac{1-\cos2x}{x^{2}}, & x 0\end{cases}$ where $\alpha,\beta\in\mathbf{R}$. If $f$ is …
  17. If $\displaystyle\lim_{x\to1}\dfrac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\dfrac{m\sqrt{5}}{n(2n)^{2/3}}$, where $\gcd(m,n)=1$, then $8m+12n$ is equal to …
  18. Let $f(x)=\displaystyle\int_{0}^{x}\left(t+\sin\left(1-e^{t}\right)\right)dt$, $x\in\mathbb{R}$. Then, $\displaystyle\lim_{x\to0}\dfrac{f(x)}{x^{3}}$ is equal to
  19. If the function $f(x)=\begin{cases}\dfrac{72^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos x}}, & x\neq0\\ a\log_e2\,\log_e3, & x=0\end{cases}$ is continuous at $x=0$, then …
  20. If the function $f(x)=\dfrac{\sin3x+\alpha\sin x-\beta\cos3x}{x^{3}}$, $x\in\mathbf{R}$, is continuous at $x=0$, then $f(0)$ is equal to :
  21. Let $f:[-1,2]\to\mathbf{R}$ be given by $f(x)=2x^{2}+x+\left[x^{2}\right]-[x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points …
  22. Let $a>0$ be a root of the equation $2x^{2}+x-2=0$. If …
  23. $\displaystyle\lim_{x\to0}\dfrac{x\left(e^{\left(\sqrt{1+x^{2}+x^{4}}-1\right)/x}-1\right)}{\sqrt{1+x^{2}+x^{4}}-1}$
  24. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}x^{5}\sin\left(\dfrac{1}{x}\right)+5x^{2}, & x 0\end{cases}$ The value of $\lambda$ for which $f''(0)$ …
  25. If $\displaystyle\lim_{x\to\infty}\left(\left(\dfrac{e}{1-e}\right)\left(\dfrac{1}{e}-\dfrac{x}{1+x}\right)\right)^{x}=\alpha$, then the value of …
  26. If the function $f(x)=\begin{cases}\dfrac{2}{x}\left\{\sin(k_1+1)x+\sin(k_2-1)x\right\}, & x 0\end{cases}$ is continuous at $x=0$, then $k_1^{2}+k_2^{2}$ is equal to :
  27. Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of the points, where the function $f(x)=[x]+|x-2|$, $-2<x<3$, is not …
  28. Let $f(x)=\displaystyle\lim_{n\to\infty}\sum_{r=0}^{n}\left(\dfrac{\tan\left(x/2^{r+1}\right)+\tan^{3}\left(x/2^{r+1}\right)}{1-\tan^{2}\left(x/2^{r+1}\right)}\right)$. …
  29. Let $[t]$ denote the greatest integer $\le t$. The number of points where the function $f(x)=[x]\left|x^2-1\right|+\sin\left(\frac{\pi}{[x]+3}\right)-[x+1]$ …
  30. Let $f:[-1,3]\to\mathbb R$ be defined as $f(x)=\begin{cases}|x|+[x], & -1\le x<1\\ x+|x|, & 1\le x<2\\ x+[x], & 2\le x\le3,\end{cases}$ where $[t]$ denotes the greatest …
  31. Let $f:\mathbb R\to\mathbb R$ be a continuous function. Then $\displaystyle\lim_{x\to\pi/4}\frac{\frac\pi4\int_2^{\sec^2x}f(x)\,dx}{x^2-\frac{\pi^2}{16}}$ is equal to:
  32. If $\displaystyle\lim_{x\to0}\dfrac{\alpha xe^{x}-\beta\log_e(1+x)+\gamma x^2e^{-x}}{x\sin^2x}=10$, $\alpha,\beta,\gamma\in\mathbf{R}$, then the value of …

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