Limits, Continuity & Differentiability · JEE Main previous year questions

2026 16 questions

1. Q192026 · 8 Apr 2026 · Shift 2

   Let $f(x)=\{\begin{array}{ll}\frac{1}{3},& x\le \pi /2\\ \frac{b(1-\sin x)}{(\pi -2x{)}^2},& x>\pi /2\end{array}$. If $f$ is continuous at $x=\pi /2$, then the value of ${\int }_0^{3b-6}|x^2+2x-3|dx$ is :

   1. A.$5$
   2. B.$2$
   3. C.$3$
   4. D.$4$
   Show answer

   Answer: D

2. Q172026 · 6 Apr 2026 · Shift 1

   The value of ${\lim }_{x\to 0}\left(\frac{x^2{\sin }^{2}x}{x^2-{\sin }^{2}x}\right)$ is:

   1. A.2
   2. B.3
   3. C.4
   4. D.6
   Show answer

   Answer: B

3. Q162026 · 6 Apr 2026 · Shift 2

   Let ${\lim }_{x\to 2}\frac{(\tan (x-2))(r{x}^2+(p-2)x-2p)}{(x-2{)}^2}=5$ for some $r,p\in \mathbf{R}$. If the set of all possible values of $q$, such that the roots of the equation $r{x}^2-px+q=0$ lie in $(0,2)$, be the interval $(\alpha ,\beta ]$, then $4(\alpha +\beta )$ equals :

   1. A.$11$
   2. B.$13$
   3. C.$17$
   4. D.$21$
   Show answer

   Answer: C

4. Q172026 · 5 Apr 2026 · Shift 1

   The product of all possible values of $\alpha$, for which $$\underset{x\to 0}{\lim }\left(\frac{1-\cos (\alpha x)\cos ((\alpha +1)x)\cos ((\alpha +2)x)}{{\sin }^2((\alpha +1)x)}\right)=2,$$ is:

   1. A.-2
   2. B.1
   3. C.-1
   4. D.$\frac{5}{4}$
   Show answer

   Answer: C

5. Q172026 · 5 Apr 2026 · Shift 2

   Let $f(x)={\lim }_{y\to 0}\frac{(1-\cos (xy))\tan (xy)}{y^3}$. Then the number of solutions of the equation $f(x)=\sin x$, $x\in \mathbb{R}$ is :

   1. A.$0$
   2. B.$2$
   3. C.$3$
   4. D.$1$
   Show answer

   Answer: C

6. Q222026 · 4 Apr 2026 · Shift 2· Numerical answer

   Let $f(x)=\{\begin{array}{ll}{e}^{x-1}& ,x<0\\ x^2-5x+6& ,x\ge 0\end{array}$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha +\beta$ is equal to ____.

   Show answer

   Answer: 4

7. Q162026 · 2 Apr 2026 · Shift 1

   If ${\lim }_{x\to 2}\frac{\sin \left(x^3-5x^2+ax+b\right)}{(\sqrt{x-1}-1){\log }_{\mathrm{e}}(x-1)}=\mathrm{m}$, then $\mathrm{a}+\mathrm{b}+\mathrm{m}$ is equal to :

   1. A.5
   2. B.6
   3. C.8
   4. D.10
   Show answer

   Answer: B

8. Q182026 · 2 Apr 2026 · Shift 2

   Let $f(x)$ be a polynomial of degree 5, and have extrema at $x=1$ and $x=-1$. If ${\lim }_{x\to 0}\left(\frac{f(x)}{x^3}\right)=-5$, then $f(2)-f(-2)$ is equal to :

   1. A.0
   2. B.50
   3. C.92
   4. D.112
   Show answer

   Answer: D

9. Q252026 · 2 Apr 2026 · Shift 2· Numerical answer

   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 discontinuous, is ____.

   Show answer

   Answer: 10

10. Q152026 · 28 Jan 2026 · Shift 1

    The value of $$\underset{x\to 0}{\lim }\frac{{\log }_e(\sec (ex)\cdot \sec (e^{2}x)\cdot \cdots \cdot \sec (e^{10}x))}{e^2-e^{2\cos x}}$$ is equal to

    1. A.$\frac{(e^{10}-1)}{2e^2(e^2-1)}$
    2. B.$\frac{(e^{20}-1)}{2e^2(e^2-1)}$
    3. C.$\frac{(e^{10}-1)}{2(e^2-1)}$
    4. D.$\frac{(e^{20}-1)}{2(e^2-1)}$
    Show answer

    Answer: D

11. Q132026 · 28 Jan 2026 · Shift 2

    Let $f(x)={\lim }_{\theta \to 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta }\right)}\sin (x-1)}{1+x^{\left(\frac{2}{\theta }\right)}(x-1)}\right),x\in \mathbb{R}$. Consider the following two statements : (I) $f(x)$ is discontinuous at $x=1$. (II) $f(x)$ is continuous at $x=-1$. Then,

    1. A.Neither (I) nor (II) is True
    2. B.Only (II) is True
    3. C.Only (I) is True
    4. D.Both (I) and (II) are True
    Show answer

    Answer: A

12. Q92026 · 24 Jan 2026 · Shift 1

    Let $\alpha ,\beta \in \mathbb{R}$ be such that the function $f(x)=\{\begin{array}{ll}2\alpha (x^2-2)+2\beta x& ,x<1\\ (\alpha +3)x+(\alpha -\beta )& ,x\ge 1\end{array}$ be differentiable at all $x\in \mathbb{R}$. Then $34(\alpha +\beta )$ is equal to

    1. A.48
    2. B.84
    3. C.36
    4. D.24
    Show answer

    Answer: A

13. Q202026 · 24 Jan 2026 · Shift 1

    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

    1. A.$\frac{2}{3}$
    2. B.$\frac{1}{2}$
    3. C.2
    4. D.$\frac{3}{2}$
    Show answer

    Answer: D

14. Q202026 · 23 Jan 2026 · Shift 1

    Let $f(x)=\{\begin{array}{ll}\frac{a{x}^2+2ax+3}{4x^2+4x-3}& ,x\ne -\frac{3}{2},\frac{1}{2}\\ b& ,x=-\frac{3}{2},\frac{1}{2}\end{array}$ be continuous at $x=-\frac{3}{2}$. If $f\circ f(x)=\frac{7}{5}$, then $x$ is equal to:

    1. A.1.4
    2. B.2
    3. C.1
    4. D.0
    Show answer

    Answer: C

15. Q172026 · 23 Jan 2026 · Shift 2

    If $f(x)=\{\begin{array}{ll}\frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x}& ,x\ne 0\\ b& ,x=0\end{array}$ is continuous at $x=0$, then $a+b$ is equal to :

    1. A.1
    2. B.2
    3. C.0
    4. D.4
    Show answer

    Answer: B

16. Q202026 · 21 Jan 2026 · Shift 1

    Let $f:\mathbf{R}\to (0,\infty )$ be a twice differentiable function such that $f(3)=18$, $f^{\prime }(3)=0$ and $f^{\prime \prime }(3)=4$. Then ${\lim }_{x\to 1}{\left({\log }_e\left(\frac{f(2+x)}{f(3)}\right)\right)}^{\frac{18}{(x-1{)}^2}}$ is equal to :

    1. A.9
    2. B.18
    3. C.1
    4. D.2
    Show answer

    Answer: D