Limits, Continuity & Differentiability Questions - JEE

Let $ f(x) = 15 - |x - 10|;\; x \in R $. Then the set of all values of x, at which the function, $ g(x) = f(f(x)) $ is not differentiable, is:

If $f(x)$ is continuous and $f\!\left(\dfrac{9}{2}\right) = \dfrac{2}{9}$, then $\lim_{x \to 0} f\!\left(\dfrac{1 - \cos 3x}{x^2}\right)$ is equal to

If $a_1 = 1$ and $a_{n+1} = \dfrac{4 + 3a_n}{3 + 2a_n}$, $n \geq 1$ and if $\lim_{n \to \infty} a_n = a$, then the value of $a$ is

If $\lim_{x \to 0} \frac{a + bx\sin x + cx\cos x}{x^4} = 2$ then $a =$ ______, $b =$ ______, $c =$ ______

If $g > p > 0$ then $\lim_{x \to \infty} \frac{dx^p + ex^{p-2} + c}{dx^q + ex^{q-2} + b}$ is

$\lim_{x \to 0} \dfrac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}$

Given that, $x^y = e^{x-y}$. Find $\dfrac{dy}{dx}$.

Find the value of \[\lim_{x \to 0} \frac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}\]

Let S be the set of all (α, β) ∈R × R such that lim x→∞ sin(x2)(loge x)α sin 1 x2 x βloge(1 + x) β = 0. Then which of the following is(are) correct?

If $y = e^{nx}$, then find the value of $\left(\frac{d^2 y}{dx^2}\right)\left(\frac{d^2 x}{dy^2}\right)$.

If $\lim_{t \to x} \dfrac{\displaystyle\int_x^t \sin^{-1}(nz)\,dz}{t^2 - x^2} = f_n(x)$, then find the value of $\lim_{x \to 0}\left([f_2(x)] + [f_4(x)]\right)$.[Note: [k] denotes greatest integer function less than or equal to k]

If $a^2 + b^2 + c^2 + ab + bc + ca \leq 0$, where $a, b, c \in R$ and $f(x) = a[x] + b|x| + c\,\text{sgn}(x)$, then in $(-2, 2)$, which of the following is not true?[Note: $[y]$ denotes greatest integer function less than or equal to $y$.]

If $f(x) = x^4 \tan x^3 - x \ln(1 + x^2)$, then the value of $\dfrac{d^4 f(x)}{dx^4}$ at $x = 0$ is:

If $f(x) = [\frac{x}{4}]$, $x \in R$, where $[x]$ denotes the greatest integer function, then :

Find $\displaystyle\lim_{x \to \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}$, where $x = \alpha$ is a root of $ax^2 + bx + c = 0$.

$\lim_{x \to 0} \frac{x^2}{y} = 0$ when $(x, y) \to (0,0)$ along the curve $y^2 = x^2$. State whether this is true or false.

Let f(x) be a function whose graph is shown. If α1, α2, α3, α4, α5 are the roots of f(x) = 0 (as shown), then the minimum number of distinct roots of $\frac{d}{dx}(f(x)f''(x)) = 0$ is:

Find $L = \lim_{x \to \infty} \dfrac{\left(\int_0^x e^{x^2}\, dx\right)^2}{\int_0^x e^{2x^2}\, dx}$.

If y = $\sqrt{x + \sqrt{y + \sqrt{x + \sqrt{y + \cdots}}}}$, then $\frac{dy}{dx}$ is equal to

If y = f(x) is an odd differentiable function defined on (−D, D) such that f'(3) = −2, then f'(−3) equals

Find $\lim_{x \to 1} \frac{-x^3 + x^2 + x - 3}{x^2 - 4x + 3}$

Let f be an even function and $f'(0)$ exists, then $f'(0)$ is

The differential coefficient of tan⁻¹((1+x²-1)/x) with respect to tan⁻¹ x, when x ≠ 0, is

Given $e^y + xy = e$. If $y = f(x)$, then find $\left(\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\right)$ at $x = 0$.

If $x^m \cdot y^n = (x+y)^{m+n}$, then $\dfrac{dy}{dx}$ is

If $\lim_{x \to \infty} \left(1 + \dfrac{a}{x} - \dfrac{4}{x^2}\right)^{2x} = e^3$, then $a$ is equal to

The value of $\lim_{x \to \infty} \frac{[1^2(\sin x)^x] + [2^2(\sin x)^x] + \ldots + [n^2(\sin x)^x]}{n^3}$ (where $[\cdot]$ denotes the greatest integer function) is

Example 45: Let $[\cdot]$ denote the greatest integer function and $f(x) = [\tan^2 x]$. Which of the following is true? [IIT JEE 1993]

If $f(x) = \text{sgn}(\cos 2x - 2\sin x + 3)$, where $\text{sgn}()$ is the signum function, then $f(x)$

If f(x) = $\frac{\cos x}{\sin x}$, then f'($\frac{\pi}{4}$) is equal to

Let $f(x) = \frac{|x^3 - 6x^2 + 11x - 6|}{x^3 - 6x^2 + 11x - 6}$, then the number of solutions of $a$, where $\lim_{x \to a} f(x)$ doesn't exist is

Assume that $f$ is continuous on $[a, b]$, $a > 0$ and differentiable on $(a, b)$. If $\dfrac{f(a)}{a} = \dfrac{f(b)}{b}$, then there exists $x_0 \in (a, b)$ such that:

The function \[f(x) = \begin{cases} \frac{|x|(3e^{1/|x|} + 4)}{2 - e^{1/|x|}} & , x \neq 0 \\ 0 & , x = 0 \end{cases}\] is

Let function f be defined as f: ℝ+ → ℝ+ and function g is defined as g: ℝ+ → ℝ+. Functions f and g are continuous in their domain. Suppose, the function h(x) = limn→∞ $\frac{f(x) + x^n}{x^n + g(x)}$, x > 0 is continuous in its domain, then f(1) × g(1) is equal to

If f''(x) = f(x), where f(x) is a continuous double differentiable function and g(x) = f'(x). If F(x) = \[8f\left(\frac{x}{2}\right) + 8g\left(\frac{x}{2}\right)\] and F(5) = 5, then F(10) is

The domain of the derivative of the functions f(x) =$$f(x) = \begin{cases} \tan^{-1} x, & \text{if } |x| > 1 \\ \frac{1}{2}(|x| - 1), & \text{if } |x| \leq 1 \end{cases}$$is

If f(x) = sin $\frac{[x]}{x^2}$ for 2 ≤ x ≤ 3 and [x] denotes the greatest integer less than or equal to x, then f'(π/3) is equal to

Let $f(x) = \frac{\sqrt{2 + \cos x}}{8x - 4\sqrt{x}}$, $g(x) = \frac{2\cos x - \sin 2x}{e^{2x} - 1}$ (where $0 + 2x$ in denominator), and $h(x) = \begin{cases} f(x) & \text{for } x \pi/2 \end{cases}$. Which of the following does not hold?

The value of $\lim_{n \to \infty} \frac{1}{n}\left([1^2 x^7 + 1^2] + [2^2 x^7 + 2^2] + \ldots + [n^2 x^7 + n^2]\right)$ (where $[\cdot]$ denotes the greatest integer function) is

If $f(x) = \begin{cases} \frac{(4x-1)^3}{x^2} \sin\left(\log\left(1 + \frac{x^2}{a}\right)\right) \left(\log\frac{x}{3}\right)^n, & x \neq 0 \\ \frac{9(\log 4)^3}{2}, & x = 0 \end{cases}$ is a continuous function at x = 0, then the value of a is equal to

Find $\lim_{b \to 0} \frac{r_{max} - r}{\sin b}$

Find the value of $\lim_{\theta \to 0} \frac{\tan(\pi \cos^2 \theta)}{\sin(2\pi \sin^2 \theta)}$

Let f and g be differentiable functions satisfying g'(a) = 2, g(a) = b and f∘g = I (Identity function). Then, f'(b) is equal to

If $x = f(t), y = y(t)$, then $\frac{d^2y}{dx^2} =$

The value of $\lim_{n \to \infty} \left(\frac{1}{na} + \frac{1}{na+1} + \frac{1}{na+2} + \ldots + \frac{1}{nb}\right)$ is

Suppose $ f(x) $ is differentiable at $ x = 1 $ and $ \lim_{h \to 0} \frac{1}{h} f(1+h) = 5 $, then $ f'(1) $ equals

lim $x \to 1$ \frac{$\sqrt{\pi}$ - $\sqrt{2}$ $\sin^{-1} x$}{$\sqrt{1 - x}$} equal to :

If $\sin y = x \sin(\alpha + y)$, then $\dfrac{dy}{dx}$ is

$\lim_{x \to 0} \dfrac{\sin(\pi \cos^2 x)}{x^2}$ is equal to

The value of $\lim_{n \to \infty} \cos^2\left(\pi\left(3n^3 + n^2 + 2n\right)\right)$ (where $n \in \mathbb{N}$):

Limits, Continuity & Differentiability Questions (1044)