Limits, Continuity & Differentiability Questions (1044)

If limx→λ 2 −λ x
λ tan(πx/2λ) = 1 e, then λ is equal to:

The function $f(x) = \frac{1 - \sin x + x}{1 - \sin x \cos x}$ is not defined at $x = \pi$. The value of $f(\pi)$, so that $f(x)$ is continuous at $x = \pi$, is

If \(\lim_{x \to 0} \left(1 + ax + bx^2\right)^{2/x} = e^3\), then

\(f(x)=\max\{x,x^2\}\). Non-diff set:

If f(x) is a polynomial of least degree such that limx→0  1 + f(x)+x2 x2 1/x = e2, then f(2) is:

The function $f(x) = [x] + \sqrt{\{x\}}$, where $[.]$ denotes the greatest integer function and $\{.\}$ denotes the fractional part function respectively, is discontinuous at

Matrix Match:(P) f diff at x=3, f'(3)=2: \(\lim_{h\to 0}\frac{f(3+h^2)-f(3-h^2)}{2h^2}\)(Q) f(-x)=f(x), f'(0) exists: f'(0)(R) \(f(x)=\frac{x}{1+e^{1/x}}\) (x≠0), 0 (x=0): Lf'(0)(S) f=max{a-x, a+x, b}, 0<a<b: non-diff points

S = non-diff of \(|2-|x-3||\). Find \(\sum_{x\in S}f(f(x))\).

The differential coefficient of $\sin^{-1}\!\dfrac{2x}{1+x^2}$ with respect to $\cos^{-1}\!\dfrac{1-x^2}{1+x^2}$ is:

If f(x) = \begin{cases} \frac{\cos x^3}{3} & ; x , then find the number of points where g(x) = f(|x|) is non-differentiable.

\(f(x)=\begin{cases}\pi/4+\tan^{-1}x & |x|\ge 1\\ \frac{1}{2}(|x|-1) & |x|. f is:

If limn→∞ √ 2n2 + n −λ √ 2n2 −n  = 1 √ 2 (where λ is a real number), then:

\(\phi(x)=[|x|-|\sin x|]\) (where [.] is GIF). Which are correct?

Let a > 0 be a root of 2x2 + x −2 = 0. If lim x→1/a 16 1 −cos(2 + x −2x2)
1 −ax2 = α + β √ 17, where α, β ∈Z, then α + β is equal to

If $y = \log_{10} x + \log_x 10 + \log_x x + \log_{10} 10$, then $\dfrac{dy}{dx}\bigg|_{x=10} =$ (answer as integer, multiply by $10\ln 10$)

Let $f(x)=x^3+3x+1$ and $g$ be its inverse. Then $g'(5)$:

Let $f(x) = \dfrac{\sin x + \cos x - 1}{x}$ for $x\neq 0$ and $f(0)=0$. If $f$ is differentiable at $x=0$, find $f'(0)$.

If $y = \log_e\!\left(\dfrac{x^2}{e^2}\right)^2$, then $\dfrac{d^2y}{dx^2}$ at $x=e$ is:

For $x > 0$, if $(2y)^{2x} = 4e^{2x-2y}$, then $\dfrac{dy}{dx}$ is equal to:

Let $f(x) = \begin{cases} -1, & -2 \le x

Given \(f(x) = \dfrac{1}{x} - \dfrac{k-1}{e^{2x}-1},\; x \neq 0\). If \(f(x)\) is continuous at \(x = 0\) and \(f(0) = 1\), find the value of \(k\).

If \(f(x) = \sin^{-1}\left(\dfrac{2 \times 3^x}{1+9^x}\right)\), then \(f'\left(-\dfrac{1}{2}\right)\) equals

Let \( f(x) \) be a non-negative continuous function such that the area bounded by the curve \( y = f(x) \), \( x \)-axis and the ordinates \( x = \dfrac{\pi}{4} \) and \( x = \beta > \dfrac{\pi}{4} \) is \( \left( \beta \sin\beta + \dfrac{\pi}{4}\cos\beta + \sqrt{2}\beta \right) \). Then \( f\!\left(\dfrac{\pi}{2}\right) \) is

If the function \[ f(x) = \begin{cases} \dfrac{\sqrt{2 + \cos x} - 1}{(\pi - x)^2}, & x \neq \pi \\ k, & x = \pi \end{cases} \] is continuous at \( x = \pi \), then \( k \) equals

If \(|f(x) - f(y)| \leq 2|x - y|^{3/2}\), then which of the following is true about \(f'(x)\)?

Let $f$ be a differentiable function in the interval $(0,\infty)$ such that $f(1)=1$ and $\displaystyle\lim_{t\to x}\frac{t^2f(x)-x^2f(t)}{t-x}=1$ for each $x>0$. Then $2f(2)+3f(3)$ is equal to

Given \(f(1) = 1\), \(f'(1) = 3\). Find the derivative of \(f[f(f(x))] + (f(x))^2\) at \(x = 1\).

If \(y = \sec(\tan^{-1} x)\), then \(\dfrac{dy}{dx}\) at \(x = 1\) is equal to:

\(\lim_{x \to 0} \dfrac{\sqrt{1 - \cos 2x}}{\sqrt{2}\, x}\) is

Let \( y \) be an implicit function of \( x \) defined by \( x^{2x} - 2x^x \cot y - 1 = 0 \). Then \( y'(1) \) equals

Let \(\alpha\) and \(\beta\) be the distinct roots of \(ax^2 + bx + c = 0\), then \(\lim_{x \to \alpha} \dfrac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}\) is equal to

Let \(f(x) = \begin{cases} (x-1)^{\frac{1}{2-x}}, & x > 1,\; x \neq 2 \\ k, & x = 2 \end{cases}\). The value of \(k\) for which \(f\) is continuous at \(x = 2\) is

For \(x \to 0\), \(\frac{\sin x}{x}\) is

If \(y = f(x)\) is differentiable for all \(x \in \mathbb{R}\), then:

If \(x^2 + y^2 = r^2\) and \(z = \dfrac{1}{r}\) then \(z = \sqrt{1 + \left(\dfrac{dy}{dx}\right)^2}\).State whether the statement is true or false.

Let \(\text{Lt}_{n \to \infty} f(n) = l\). Given that \(\text{Lt}_{n \to \infty} f(n+1) = \dfrac{1}{2} \text{Lt}_{n \to \infty} \left[f(n) + \dfrac{9}{f(n)}\right]\) and \(f(n) > 0 \; \forall n \in N\), find \(\text{Lt}_{n \to \infty} f(n)\).

It is given that \(f(x) = \text{Min}\{x+1, |x|+1\}\). Then which of the following is true?

Let BC be the diameter of a circle centred at O. Point A is variable on circumference. If BC = 1, then limA→B BM (Area of sector OAB)2 is:

\(\lim_{x\to\infty} \dfrac{\left(\displaystyle\int_0^x e^{x^2}\,dx\right)^2}{\displaystyle\int_0^x e^{2x^2}\,dx} = \)

The value of a for which \(\lim_{x \to 0} \dfrac{\left(e^x - 1\right)^4}{\sin\!\left(\dfrac{x^2}{a^2}\right)\ln\!\left(1 + \dfrac{x^2}{2}\right)} = 8\) is

\(\lim_{x \to 0} \dfrac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}\) is equal to: (where \([\,]\) denotes greatest integer function)

Evaluate: \(\lim_{n \to \infty} \dfrac{(3n+3)(3n+2)(3n+1)(3n!)}{(n+1)^3 (3n)!}\)

Evaluate \(\lim_{x \to \infty} \dfrac{x^a \sin\dfrac{1}{x^a} + x^2}{1 + |x|^3}\) when \(x \to \infty\).

Evaluate \(\lim_{x \to \pi/2} \tan x \cdot \log \sin x\).