Differentiability Questions (1063)

If $$I_1 = \lim_{x \to \infty} (\tan^{-1}(-x) - \tan^{-1}x)\cos x$$ and $$I_2 = \lim_{x \to 0} (\tan^{-1}(-x) - \tan^{-1}x)\cos x$$, then $$(I_1, I_2)$$ is

Let \(f(x) = x^{\alpha - 1} \cdot \dfrac{e^{2/x} - 1}{e^{2/x} + 1}\) for \(x \neq 0\) and \(f(0) = 0\). For \(f\) to be differentiable at \(x = 0\), the value of \(\alpha\) must satisfy:

Find $$\lim_{n \to \infty} \frac{1^4 + 2^4 + 3^4 + \ldots + n^4}{n^5} - \lim_{n \to \infty} \frac{1^3 + 2^3 + 3^3 + \ldots + n^3}{n^5}$$

Given \[ f(x) = \begin{cases} |x| + [x], & -1 \leq x Find the number of points where \(f(x)\) is discontinuous.

Evaluate \(\lim_{x \to 0^+} (\sin x)^{\tan x}\)

If limx→0(1 + ax + bx2)2/x = e3, then:

Let $f(x) = \dfrac{(2^x+2^{-x})\tan x\sqrt{\tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^3}$. Then $f'(0)$ is:

Let $f(x) = \begin{cases} a & x = \frac{2}{3} \\ \frac{\sqrt{9x+3}-\sqrt{3}}{\sqrt{9-7x+4}} & x > \frac{2}{3} \end{cases}$ If $f(x)$ is continuous at $x = \frac{2}{3}$, then the value of $\frac{a}{b}$ is

[Bonus/Extra question from JEE Main 2022 set — verify in source] If $2\sin\theta = e^x + e^{-x}$, find $\dfrac{d^2y}{dx^2} + n^2y = 0$ type ODE verification. [Intentionally left for manual verification]

lim n→∞ (12 −1)(n −1) + (22 −2)(n −2) + · · · + ((n −1)2 −(n −1)) · 1 (13 + 23 + · · · + n3) −(12 + 22 + · · · + n2) is equal to

If f: \mathbb{R} \to \mathbb{R}\) is a function defined by f(x) = [x]\cos\left(\frac{2x-1}{2}\pi\right)\), where [x]\) denotes the greatest integer function, then f is

The value of f at x = 0, so that function f(x) = \frac{2^x - 2^{-x}}{x}\), x \neq 0\) is continuous at x = 0, is

If $$f(x) = 0$$ is a quadratic equation such that $$f(-\pi) = f(\pi) = 0$$ and $$f\left(\frac{3\pi}{2}\right) = -\frac{3\pi}{4}$$, then $$\lim_{x \to -\pi} \frac{f(x)}{\sin(\sin x)}$$ is equal to

If f : ℝ → ℝ is the function defined by f(x) = (ex − e−x)/(ex + e−x), then

Let \(f:[−1,3]\to\mathbb{R}\) be defined as \(f(x)=\begin{cases}|x|+[x], & -1\leq x where \([t]\) denotes the greatest integer less than or equal to \(t\). Then \(f\) is discontinuous at

Let \(f(x) = \int_0^x t\ln(1+t^2)\, dt\), then \(f''(0)\) is:

If $2^x + 2^y = 2^{x+y}$, then $\dfrac{dy}{dx}$ is equal to which of the following expressions?

Let f : R → R be a continuous function satisfying \( f(x) + \int_0^x t f(t)\, dt + x^2 = 0 \) ∀ x. Then:

Let \(h(x) = \min\{x, x^2\}\) for every real number \(x\). Then,

Let \[S = \{(\lambda, \mu) \in R \times R : f(t) = (|\lambda|e^{|t|} - \mu)\sin(2|t|),\, t \in R,\] is a differentiable function\}. Then \(S\) is a subset of

If $f(x) = \begin{cases} 2x^2 + 3 & x > 3 \\ ax^2 + bx + 1 & x \leq 3 \end{cases}$ is differentiable everywhere, then $a$ is equal to

252. The number of points where \(f(x) = |x + [x]| - 3[2x] + 4[3x]\) is discontinuous in \([-1, 1]\), is: [Note: \([k]\) denotes greatest integer less than or equal to \(k\).]

The function $f(x) = \max\{(1-x), (1+x), 2\}$ $\forall x \in \mathbb{R}$ is

If $y = \tan^{-1}\left(\frac{1}{1+x^2}\right) + \tan^{-1}\left(\frac{2x}{1-x^2}\right)$ ($y > 0$), then $\frac{dy}{dx}$ is equal to

If $a = \sec f$ and $b = \csc f$, where $f$ is a parameter, then the value of $\frac{dy}{dx}$ when $x = -\frac{3}{4}$ is

If $y = (1+x^2)^n + \sin^{-1}(\sin^2 x)$, then $\frac{d^2y}{dx^2}$ at $x = 0$ is

Let \(f(x) = x\left[\frac{x}{2}\right]\), for \(-10

Let h(x) = 2 − |x − 1| and g(x) = h(|x|) + |h(x)|. Find the number of points where g(x) is non-differentiable.

The value of $\lim_{x \to \infty} \frac{ax^2 + bx + c}{dx + e}$ ($a, b, c, d, e \in \mathbb{R} - \{0\}$) depends on the sign of:

If $f(x) = 2x^3 - 3x^2 + a$ is a one-to-one function, the value of $a$ is arbitrary. Given $f(g(x))=x$, then $g'\!\left(\dfrac{11}{4}\right)$ is:

\[ \lim_{x \to 0} \frac{\sin{(\pi \cos{x})}}{x^2} \]

Find $\frac{dy}{dx}$ at $x = 3$ where $y = \sqrt{x^2 + 16}$ and $s = \frac{x}{y}$

If $x = \frac{1-\tan^2\theta}{1+\tan^2\theta}$ and $y = \frac{2\tan\theta}{1+\tan^2\theta}$, find $\frac{dy}{dx}$

25. If \(y = (x + \sqrt{1 + x^2})^n\), then the value of \((1 + x^2)\dfrac{d^2y}{dx^2} + x\dfrac{dy}{dx}\) is equal to:

Consider the function $f(x) = \tan^{-1}\left(\frac{x^2-1}{x^2+1}\right), \forall x \geq 0$. If $g(x)$ is the inverse function of $f(x)$, then the value of $g'\left(\frac{\pi}{4}\right)$ is equal to

The integer \(n\) for which \(\lim_{x \to 0} (\sin x)^{1/x}\) is a finite non-zero number, is [2002 AIEEE]

If $I_n = \frac{d^n}{dx^n}(x^n\ln x)$, then the value of $\frac{d}{dx}(I_7 - 7I_6)$ is equal to

198. If \(y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}\), then the value of \(\dfrac{dy}{dx}\) at \(x = 8\) is:

The total number of points of non-differentiability of \(f(x) = \max\left\{\sin^2 x, \cos^2 x, \frac{3}{4}\right\}\) in \([0, 10\pi]\), is

If $\lim_{x \to \frac{\pi}{2}} \frac{\sin x}{x}$ exists finitely, then the value of $a$ is

The value of $\lim_{x \to \pi} \left[1 - \cos\left(\frac{x}{2}\right) - \cos\left(\frac{x}{4}\right) + \cos\left(\frac{x}{2}\right) \cdot \cos\left(\frac{x}{4}\right)\right]$ is $\frac{\lambda}{4}$, then the value of $900\lambda$ is equal to there, $\lambda > 0$)

If $\lim_{x \to 0} \frac{\sin 3x}{\sin 4x} - 3$, then the value of $272 \cdot \frac{ab}{cd}$ is equal to

The value of $\lim_{x \to 0} \frac{\sin\left(\frac{x}{3}\right) \sin\left(\frac{x}{3}\right)}{\left(\frac{x}{3}\right)}$ is equal to

If $x = t^3 + t^5$ and $y = \sin t$, then $\dfrac{d^2y}{dx^2}$ equals:

Let $f(x) = x^2 - 4x - 3, x > 2$ and $g(x)$ is the inverse of $f(x)$. Then the value of $\frac{1}{g'(2)}$, where $f(z) = 2$ is (here, $g'$ represents the first derivative of $g$)

Find \(\displaystyle\lim_{x \to \infty} \frac{\displaystyle\int_0^x e^{t^2}\, dt}{\displaystyle\int_0^x e^{2t^2}\, dt}\).

198. If \(y = \sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\dfrac{x}{\sqrt{\cdots}}}}}}}\), then the value of \(\dfrac{dy}{dx}\) at \(x = 8\) is:

Find $\lim_{x \to 4} \frac{(1-\sin(\pi x))}{(x-4)^2}$

If \(y = 1 + \dfrac{c_1}{x - c_1} + \dfrac{c_2 x}{(x - c_1)(x - c_2)} + \dfrac{c_3 x^2}{(x - c_1)(x - c_2)(x - c_3)}\), then \(\dfrac{dy}{dx}\) is equal to:

Let \(f\) be continuous on \([a, b]\), differentiable on \((a, b)\), and \(f(a)/a = f(b)/b\). Then by Rolle's theorem applied to \(g(x) = f(x)/x\), there exists \(x_0 \in (a, b)\) such that: