Limits, Continuity & Differentiability Questions (1044)

Let y = 2x tan−1x − ln(1 + x2). Find the number of values taken by 5 − |[x]|.

Evaluate \(\lim_{x \to \infty} \left(\sqrt{x + \sqrt{x + \sqrt{x}}} - \sqrt{x}\right)\)

The graph of the function \(f(x) = \cos x \cdot \cos(x+2) - \cos^2(x+1)\) is

The value of $\lim_{x \to 0} \frac{\sin(\tan x)}{x(2 + \cos(3x))}$ is equal to

If \(\lim_{x \to 1}\left(2 - x + a[x-1] + b[1+x]\right)\) exists, then a and b can take values (where [.] denotes greatest integer function)

If \(\lim_{x \to 2} \dfrac{\tan(x-2)}{x-2} \cdot \dfrac{x^2+kx-2x-2k}{(2-k)} = 5\), find \(k\).

If $f(x) = x + 2$, then $f'(f(x))$ at $x = 4$ is:

The value of $\lim_{x \to 0} \frac{(1 + \sin 2x)^{1/\cos x}}{x \sin 8x}$ is equal to

Select correct statements:

\(\lim_{x \to \pi/2} \frac{2x - \pi}{\cos x}\) is equal to

If \[f(x) = \begin{cases} \dfrac{\sin(p+1)x + \sin x}{x}, & x 0 \end{cases}\] is continuous at \(x = 0\), then the ordered pair \((p,\,q)\) is equal to:

Evaluate: \(\lim_{x \to 0} \dfrac{x^2 \sin\left(\dfrac{1}{x}\right) + 2x}{x} \cdot \dfrac{x}{(1+x)^{1/x} - e}\)

If \(f(x)=(x^5+1)|x^2-4x-5|+\sin|x|+\cos(|x-1|)\), then \(f(x)\) is NOT differentiable at:

Evaluate: \(\displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{\cot^3 x - \tan x}{\cos\left(x + \dfrac{\pi}{4}\right)}\)

If \(f(x) = \begin{cases} \left(\left(\sin\left(\dfrac{2x^2}{a}\right) + \cos\left(\dfrac{3x}{b}\right)\right)^{\frac{ab}{x^2}}, & x \neq 0 \\ e^{x^2 - 2x + 3}, & x = 0 \end{cases}\) is continuous at \(x = 0\), where \(b \in R\), then the minimum value of \(a\) is:

If \( f(x) = \begin{cases} xe^{-\left(\frac{1}{|x|}+\frac{1}{x}\right)}, & x \neq 0 \\ 0, & x = 0 \end{cases} \) then \( f(x) \) is

At $x = \dfrac{\pi}{4}$, $\dfrac{d}{dx}\!\left(\sin(\sin x)\right)$ equals:

Evaluate $\lim_{n \to \infty} \frac{\sin(\sqrt{n})-\sin\sqrt{n-1}}{n^1}$

$\lim_{x \to 0} \frac{x^2 + 5x}{x^2 + x^3}$

If f is a differentiable function and \[\lim_{x \to 1} \frac{f(1+x^3-x)-f(x)}{\sin(x-1)}\] exists finitely, find \(f'(1)\).

If \(f(0) = 1\) and \(\displaystyle\lim_{t \to x} \frac{\sec x \cdot f(t) - f(x) \sec t}{t - 1} = \sec^2 x\). The value of \(\dfrac{f(0)}{f'(0)}\), is:

Let \(f(x) = |x - \pi| \cdot (e^{|x|} - 1)\sin|x|\)Then which of the following is true?(The set S of points where f is not differentiable)

\(f(x)=(x^2-1)|x^2-3x+2|+\cos(|x|)\) is NOT differentiable at:

If \[ \lim_{\alpha \to 0} \frac{e^{\cos(\alpha^n)-1}}{\alpha^m} = -\frac{1}{2} \] then find the value of \(m - 2n\).

Let $$f(x)$$ be a real valued function defined for all $$x \neq -1, 1$$, satisfying $$f(1) = 1$$ and $$f'(x) = \frac{1}{x^2 + (f(x))^2}$$; then $$\lim_{x \to \infty} f(x)$$

If 0 and f(a, b) = \frac{\tan b - \tan a}{b - a}, then

Find \lim_{x \to 2} \frac{x - 2}{|x - 2|}\

If f(x) = \begin{cases} [x]-[-x] & , x \neq 2 \\ ; & , x = 2 \end{cases} and f is continuous at x=2, where [\cdot] denotes greatest integer function, then ; is

Let f(x) satisfy the requirements of Lagrange's mean value theorem in [0, 2]. If f(0) = 0 and |f'(x)| \leq \frac{1}{2} for all x \in [0, 2], then

Let f(x) be a twice differentiable function defined on (-\infty, \infty) such that f(x) = f(2-x) and f'\left(\frac{1}{2}\right) = f'\left(\frac{1}{4}\right) = 0.The minimum number of values where f''(x) vanishes on [0, 2] is:

Let [ ]x denote the greatest integer less than or equal to x. Then, \lim_{x \to 0} \frac{\tan(\pi \sin^2 x) + (|x| - \sin(x[x]))^2}{x^2}

If y = (\cos x)^{(\cos x)^{(\cos x)^{\cdots \infty}}}, then \frac{dy}{dx} is equal to

Find the number of points where f(x) = [\sin x - \cos x] (where [\cdot] denotes greatest integral function), x \in [0, 2\pi] is not continuous.

\lim_{n \to 0} \left(1 - \frac{1}{2^2}\right) \left(1 - \frac{1}{3^2}\right) \cdots \left(1 - \frac{1}{n^2}\right) is equal to

161. \(\lim_{x \to \infty} x\left(\left(\dfrac{x}{x+1}\right)^x - \dfrac{1}{e}\right)\) is equal to:

If y = e^{2\sin^{-1}x}, then find \frac{(x^2-1)y'' + xy'}{y}.

Let f: \mathbb{R} \to \mathbb{R} is a function satisfying f(x + y^3) = f(x) + [f(y)]^3 for all x, y \in \mathbb{R}. If f'(0) \geq 0, then f(10) is

If f(x) = \begin{cases} \frac{\sin^3(3x) \times \log(1+3x)}{(\tan^{-1}x)^2(e^{5x}-1)x}, & x \neq 0 \\ a, & x = 0 \end{cases}\) is continuous in [0,1]\), then a equals

If $h(x) = \begin{cases} \frac{\lambda\sqrt{2x+3}}{0 \leq x \leq 3} \\ \mu x + 12, & 3 < x \leq 9 \end{cases}$ is differentiable at $x = 3$, then the value of $\lambda + \mu$ is equal to

If \lim_{n \to \infty} (\sin^{-1} x)^{n+1} = 1, find the interval in which x lies.

If $f(x) = \begin{cases} \frac{\sqrt{x+3}-2}{x-1} & 0 \leq x < 4 \\ b & x \geq 4 \end{cases}$ is continuous at $x = 4$, then value of $\frac{a}{b}$ is equal to

Let f(x) = \begin{cases} \tan\left(\frac{\pi}{4} + x\right)^{1/x} & , x \neq 0 \\ k & , x = 0 \end{cases}For what value of k is f(x) continuous at x = 0?

If $f(x) = \begin{cases} px + q & x \leq 2 \\ x^2 - 5x + 6 & 2 < x < 3 \\ a x^2 + bx + 1 & x \geq 3 \end{cases}$ is differentiable everywhere, then $|p| + |q| + \left|\frac{1}{3}\right| + |1|$ is equal to

Let \(f(x) = \begin{cases} x + 3 & -2

g defined via running max. Non-differentiable points in (0,4):

If $\log_{10}\left(\frac{x^2}{y}\right) = 2$, then $\frac{dy}{dx} =$

\(\lim_{x \to 0} \dfrac{x \tan 2x - 2x \tan x}{(1 - \cos 2x)^2}\) equals

If $f(1) = 1,\; f'(1) = 3$, then the derivative of $f(f(f(x))) + (f(f(x)))^2$ at $x = 1$ is:

Let \(f(x) = \begin{cases} \sqrt{x^2 - 1}, & x \leq \sqrt{10} \\ (\sqrt{10}x - 7), & \sqrt{10}

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