JEE Limits, Continuity & Differentiability PYQs with PDF

Let $$f: R \rightarrow (0, \infty)$$ be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f" (3) = 4. Then $$\lim_{x \rightarrow 1}\left(\log_{a}\left(\frac{f(2+x)}{f(3)}\right)^{\frac{18}{(x-1)^{2}}}\right)$$ ls equal to :

If $$\lim_{x \rightarrow 0} \frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}}{x \cos x-\log_{e}{(1+x)}} =2$$, then $$a^{2}+b^{2}+c^{2}$$ is equal to :

Lety = y (x) be a differentiable function in the interval $$(0, \infty)$$ such that y(l) = 2, and $$\lim_{t \rightarrow x} \left( \frac{t^{2}y(x)-x^{2}y(t)}{x-t} \right) = 3$$ for each x > 0. Then 2){2) is equal to

The value of $$\lim_{x \to 0}\frac{\log_e\!\left(\sec(ex)\cdot \sec(e^{2}x)\cdots \sec(e^{10}x)\right)}{e^{2}-e^{2\cos x}}$$ is equal to

Let $$f(x) = \lim_{\theta \to 0}\left(\frac{\cos\pi x - x^{\frac{2}{\theta}} \sin(x - 1)}{1 + x^{\left(\frac{2}{\theta}\right)} (x - 1)}\right), \quad x \in \mathbb{R}$$. Consider the following two statements :

(I) $$f(x)$$ is discontinuous at $$x=1$$.
(II) $$f(x)$$ is continuous at $$x= - 1$$.
Then,

If $$\lim_{x \rightarrow \infty}((\frac{e}{1-e})(\frac{1}{e}-\frac{x}{1+x}))^{x}=\alpha$$ then the value of $$\frac{\log_{e}^{\alpha}}{1+\log_{e}^{\alpha}}$$ equals :

$$\lim_{x \rightarrow \infty}\frac{(2x^{2}-3x+5)(3x-1)^{\frac{x}{2}}}{(3x^{2}+5x+4)\sqrt{(3x+2)^{x}}}$$ is equals to :

$$ \text{Let } f:\mathbb{R}\setminus\{0\}\to\mathbb{R} \text{ be a function such that } f(x)-6f\!\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}. \text{ If } \lim_{x\to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta, \; \alpha,\beta\in\mathbb{R}, \text{ then } \alpha+2\beta \text{ is equal to:} $$

$$\lim_{x\to 0}\cosec x \left( \sqrt{2\cos^2 x+3\cos x} - \sqrt{\cos^2 x+\sin x+4} \right)$$ is:

Let [t] be the greatest integer less than or equal to t. Then the least value of $$p \in N$$ for which $$\lim_{x\rightarrow 0^{+}}\left(x([\frac{1}{x}]+[\frac{2}{x}]+...+[\frac{p}{x}])-x^{2}([\frac{1}{x^{2}}]+[\frac{2^{2}}{x^{2}}]+...+[\frac{9^{2}}{x^{2}}])\right) \geq 1$$ is equal to_______.

If $$\lim_{x\to 1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}}$$, where gcd(m, n) = 1, then $$8m + 12n$$ is equal to ______.

Let $$f(x) = \lim_{n \to \infty} \sum_{r=0}^{n}\left(\frac{\tan\left(\frac{x}{2^{r+1}}\right) + \tan^{3}\left(\frac{x}{2^{r+1}}\right)}{1 - \tan^{2}\left(\frac{x}{2^{r+1}}\right)}\right).\quad$$ Then $$\lim_{x \to 0} \frac{e^{x} - e^{f(x)}}{x - f(x)}$$ is equal to.

If $$a = \lim_{x \to 0} \frac{\sqrt{1 + \sqrt{1 + x^4}} - \sqrt{2}}{x^4}$$ and $$b = \lim_{x \to 0} \frac{\sin^2 x}{\sqrt{2} - \sqrt{1 + \cos x}}$$, then the value of $$ab^3$$ is :

If $$\lim_{x \to 0} \frac{3 + \alpha \sin x + \beta \cos x + \log_e(1 - x)}{3\tan^2 x} = \frac{1}{3}$$, then $$2\alpha - \beta$$ is equal to :

$$\lim_{x \to \frac{\pi}{2}} \left(\frac{1}{(x - \frac{\pi}{2})^2} \int_{x^3}^{(\frac{\pi}{2})^3} \cos\left(\frac{1}{t^3}\right) dt\right)$$ is equal to

Let $$f(x) = \sqrt{\lim_{r \to x}\left\{\frac{2r^2[(f(r))^2 - f(x)f(r)]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}}\right\}}$$ be differentiable in $$(-\infty, 0) \cup (0, \infty)$$ and $$f(1) = 1$$. Then the value of $$ae$$, such that $$f(a) = 0$$, is equal to ______.

If the domain of the function $$f(x) = \cos^{-1}\left(\frac{2 - |x|}{4}\right) + (\log_e(3 - x))^{-1}$$ is $$[-\alpha, \beta) - \{\gamma\}$$, then $$\alpha + \beta + \gamma$$ is equal to :

If the domain of the function $$f(x) = \log_e\frac{2x+3}{4x^2+x-3} + \cos^{-1}\frac{2x-1}{x+2}$$ is $$(\alpha, \beta]$$, then the value of $$5\beta - 4\alpha$$ is equal to

$$\lim_{x \to 0} \frac{e^{2\sin x} - 2\sin x - 1}{x^2}$$

Let $$f: \mathbb{R} \rightarrow (0, \infty)$$ be strictly increasing function such that $$\lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1$$. Then, the value of $$\lim_{x \to \infty} \left[\frac{f(5x)}{f(x)} - 1\right]$$ is equal to

If $$\lim_{x \to 0} \frac{ax^2e^x - b\log_e(1+x) + cxe^{-x}}{x^2\sin x} = 1$$, then $$16(a^2 + b^2 + c^2)$$ is equal to

Let $$f(x) = \int_0^x (t + \sin(1 - e^t))dt$$, $$x \in \mathbb{R}$$. Then, $$\lim_{x \to 0} \frac{f(x)}{x^3}$$ is equal to

If the function $$f(x) = \begin{cases} \frac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_e 2 \log_e 3, & x = 0 \end{cases}$$ is continuous at $$x = 0$$, then the value of $$a^2$$ is equal to

Let $$a > 0$$ be a root of the equation $$2x^2 + x - 2 = 0$$. If $$\lim_{x \to \frac{1}{a}} \frac{16(1 - \cos(2 + x - 2x^2))}{(1 - ax)^2} = \alpha + \beta\sqrt{17}$$, where $$\alpha, \beta \in Z$$, then $$\alpha + \beta$$ is equal to ______

If the function $$f(x) = \frac{\sin 3x + \alpha \sin x - \beta \cos 3x}{x^3}, x \in \mathbb{R}$$, is continuous at $$x = 0$$, then $$f(0)$$ is equal to :

Let $$f: R\rightarrow R$$ be a twice differentiable function such that $$f''(x) > 0$$ for all $$x\in R$$ and f'(a-1)=0, where a is a real number. Let g(x)= $$f(\tan^{2}x- 2\tan x+a)$$, $$0 < x < \frac{\pi}{2}$$.
Consider the following two statements :
(I) $$\text{g is increasing in } \left(0, \frac{\pi}{4} \right)$$
(II) $$\text{g is deceasing in } \left( \frac{\pi}{4} , \frac{\pi}{2} \right)$$
Then,

Let $$f(x) = x^{3}+ x^{2}f'(1)+2xf''(2)+f'''(3)$$, $$x\epsilon R$$. Then the value of f'(5) is :

Let $$\left[\cdot\right]$$ denote the greatest integer function, and let f (x) = $$\min \left\{\sqrt{2x},x^{2}\right\}$$. Let S = $$\left\{x \in (-2,2): \text{the function,} g(x)= |x|\left[x^{2}\right]\text{is discontinuous at x} \right\}.$$ Then $$\sum_{x\in S}f(x)$$ equals

Let $$f(x) = \left\{\begin{array}{l l}\frac{ax^{2}+2ax+3}{4x^{2}+4x-3} ,& x\neq\quad -\frac{3}{2},\frac{1}{2}\\b, & \quad x=-\frac{3}{2},\frac{1}{2}\\\end{array}\right.$$

be continuous at $$x=-\frac{3}{2}$$. If $$fof(x)=\frac{7}{5}$$ then x is equal to:

Let $$ f(x)=x^{2025}-x^{2000}, x \text{ }\epsilon \text{ }[0,1] $$ and the minimmu value of the function $$ f(x)$$ in the interval [0, 1] be $$(80)^{80}(n)^{-81}$$. Then n is equal to

If the function $$f(x)=\frac{e^{x}(e^{\tan x-x}-1)+\log_{e}{(\sec  x+\tan x)}-x}{\tan x-x}$$ is continuous at x = 0, then the value of f(O) is equal to

Let $$\alpha, \beta \epsilon \mathbb R$$ be such that the fonction $$f(x) =\left\{\begin{array}{||}2\alpha(x^{2}-2)+2\beta x & \quad,{x<1}\\(\alpha +3)x+(\alpha -\beta) & \quad ,{x\geq 1}\\\end{array}\right.$$ be differentiable at all $$x\epsilon \beta$$. Then $$34(\alpha +\mathbb R)$$ is equal to

Let $$(2a, a)$$ be the largest interval in which the function $$f(t)=\frac{|t+1|}{t^{2}},t < 0$$, is strictly decreasing. Then the local maximum value of the function $$g(x)=2\log_{e}(x-2)+a x^{2}+4x-a,x > 2$$, is______.

Let [t] denote the greatest integer less than or equal to t. If the function $$f(x) = \begin{cases} b^2 \sin\!\left(\dfrac{\pi}{2}\left[\dfrac{\pi}{2}(\cos x + \sin x)\cos x\right]\right), & x < 0 \\[10pt] \dfrac{\sin x - \dfrac{1}{2}\sin 2x}{x^3}, & x > 0 \\[10pt] a, & x = 0 \end{cases}$$ is continuous at x = 0,then $$a^{2} + b^{2}$$ is equal to

Consider the following three statements for the function $$f: (0, \infty ) \rightarrow \mathbb R$$ defined by
$$f(x)= |\log_{e}{x}|-|x-1|:$$
(I)f is differentiable at all x > 0.
(II)f is increasing in (0, 1).
(III)f is decreasing in (1, $$\infty$$).
Then.

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

The sum of all the elements in the range of $$f(x) =Sgn(\sin x) + Sgn(\cos x) +Sgn(\tan x) +Sg n(\cot x)$$, $$x \neq \frac{n\pi}{2}, n\epsilon Z, \text{ where } Sgn(t)=\begin{cases}1, & \text{ if } t>0\\-1 & \text{ if } t<0\end{cases} ,is:$$

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 continuous and not differentiable. Then  $$m+n$$  is equal to:

Let $$f(x) = \left\{\begin{array}{l l}3x & \quad {x<0}\\min\left\{1+x+[x],x+2[x]\right\}, & \quad {0\leq x\leq 2}\\ 5, & \quad {x>2,} \end{array}\right.$$ where [.] denotes greatest integer function. If $$\alpha$$ and $$\beta$$ are the number of points, where f is not continuous and is not differentiable, respectively, then $$\alpha +\beta$$ equals_________

If the function $$f(x)=\begin{cases}\frac{2}{x}\{\sin((k_1+1)x)+\sin(k_2-1)x\}, & x<0 \\4, & x=0 \\\frac{2}{x}\log_e\left(\frac{2+k_1x}{2+k_2x}\right), & x>0\end{cases}$$ is continuous at x=0, then $$k_1^2+k_2^2$$ is equal to

Let the function f(x) = $$(x^{2}+1) |x^{2}-ax+2|+\cos|x|$$ be not differentiable at the two points x = $$\alpha$$ = 2 and $$x= \beta$$. Then the distance of the point $$(\alpha , \beta)$$ from the line $$12x+5y+10=0$$ is equal to :

Let $$f : R to R$$ be a function given by $$f(x) = \begin{cases}\frac{1-\cos 2x}{x^2}, & x < 0\\ \alpha, & x = 0\\ \frac{\beta\sqrt{1-\cos x}}{x}, & x > 0\end{cases}$$, where $$\alpha, \beta \in R$$. If f is continuous at x = 0, then $$\alpha^2 + \beta^2$$ is equal to:

Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} \frac{a - b\cos 2x}{x^2}, & x < 0 \\ x^2 + cx + 2, & 0 \leq x \leq 1 \\ 2x + 1, & x > 1 \end{cases}\\$$ If $$f$$ is continuous everywhere in $$\mathbb{R}$$ and $$m$$ is the number of points where $$f$$ is NOT differentiable then $$m + a + b + c$$ equals:

Let $$f(x) = |2x^2 + 5|x| - 3|$$, $$x \in R$$. If $$m$$ and $$n$$ denote the number of points where $$f$$ is not continuous and not differentiable respectively, then $$m + n$$ is equal to:

Consider the function $$f(x) = \begin{cases} \frac{a(7x - 12 - x^2)}{b|x^2 - 7x + 12|}, & x < 3 \\ 2^{\frac{\sin(x-3)}{x - [x]}}, & x > 3 \\ b, & x = 3 \end{cases}$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$. If $$S$$ denotes the set of all ordered pairs $$(a, b)$$ such that $$f(x)$$ is continuous at $$x = 3$$, then the number of elements in $$S$$ is :

Consider the function $$f: (0, 2) \to \mathbb{R}$$ defined by $$f(x) = \frac{x}{2} + \frac{2}{x}$$ and the function $$g(x)$$ defined by $$g(x) = \begin{cases} \min\{f(t)\},\ 0 < t \leq x & \text{and } 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{cases}$$. Then

If the function $$f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases}$$ is differentiable on $$\mathbb{R}$$, then $$48(a + b)$$ is equal to _______.

Let $$a$$ and $$b$$ be real constants such that the function $$f$$ defined by $$f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 \\ bx + 2, & x > 1 \end{cases}$$ be differentiable on $$R$$. Then, the value of $$\int_{-2}^{2} f(x) \, dx$$ equals

Let $$g(x)$$ be a linear function and $$f(x) = \begin{cases} g(x), & x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{1/x}, & x > 0 \end{cases}$$, is continuous at $$x = 0$$. If $$f'(1) = f(-1)$$, then the value of $$g(3)$$ is

Consider the function $$f: (0, \infty) \rightarrow R$$ defined by $$f(x) = e^{-|\log_e x|}$$. If $$m$$ and $$n$$ be respectively the number of points at which $$f$$ is not continuous and $$f$$ is not differentiable, then $$m + n$$ is