Limits Questions (1092)

The value of \(\lim_{x \to 0^+} x^m (\log x)^n\), \(m, n \in \mathbb{N}\) is

Given $f(x) = \begin{cases} x^2e^{-x} & 0 \leq x \leq 1 \\ a \sin(x+1) \cos(2x-2) + bx^3 & 1 < x \leq 2 \end{cases}$. If $f(x)$ is differentiable at $x = 1$, then the value of $|a - b|$ is

The value of \(\lim_{x \to 0}(f(x) + g(x) + 3)^{1/x}\) equal to:

Let \(f(x) = \left(\dfrac{4}{5}\right)^{\frac{\tan 4x}{\tan 5x}}\). If \(\displaystyle\lim_{x \to \pi/2} f(x) = k + \dfrac{2}{5}\), find the value of \(k\).

If $f(x) = \cos x\cdot\cos 2x\cdot\cos 4x\cdot\cos 8x\cdot\cos 16x$, then $f'\!\left(\dfrac{\pi}{4}\right)$ equals:

Let f(x) = x|x|, g(x) = sin x and h(x) = (g ∘ f)(x). Then

If \(\lim_{x \to 0} f(x)\) and \(\lim_{x \to 0} g(x)\) exist then \(\lim_{x \to 0} g(x)\) exist. \(\lim_{x \to 0} g(x)\) exist.

Let $f(x) = \sin\!\left(\sin^{-1}(2x)+2\tan^{-1}(2x)\right)$. If $3f'(0)=$ (integer), find it.

Let \(f(x)=x\), \(g(x)=|1-f(x)|\), \(h(x)=2-g(x)\), \(L(x)=h(|x|)+|h(x)|\). Find the number of points where \(L(x)\) is non-differentiable.

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

If \(x\log_e(\log_e x) - x^2 + y^2 = 4\), then \(\dfrac{dy}{dx}\) at \(x = e\) is equal to:

If \(f(x)\) is a differentiable function in the interval \((0, \infty)\) such that \(f(1) = 1\) and \(\lim_{t \to x} \dfrac{t^2 f(x) - x^2 f(t)}{t - x} = 1\), for each \(x > 0\), then \(f(3/2)\) is equal to

State whether the statement is true or false: \(\lim_{y \to 0} \frac{\log(1+x)}{3^y - 1} = \frac{1}{\log_e 3}\)

Let \(f\) and \(g\) be inverse functions of each other. If \(f(1) = 3\) and \(f'(1) = 4\), then \(g'(3)\) equals:

If \(\lim_{x \to 0} \frac{1-\cos 2x \sin 5x}{x^2 \sin 3x} = \frac{10}{3}\)State whether this statement is true or false.

\(\lim_{x \to 0} \dfrac{(2x)^4\left(\dfrac{1-\cos 2x}{4x^2}\right)^2}{2x\left(\tan x - \dfrac{\tan 2x}{2}\right)}\)

Consider the function \( f(x) = |x-2| + |x-5|, \, x \in R \).Statement-1: \( f'(4) = 0 \).Statement-2: \( f \) is continuous in \([2, 5]\), differentiable in \((2, 5)\) and \( f(2) = f(5) \).

If \(f(x) = \tan^2\left[x - \frac{\pi}{4}\right]\) where \([x]\) is the greatest integer function then:

If \(|f(x)| \leq x^2\), then which of the following is true?

Let \( f: R \to R \) be a function defined by \( f(x) = \min\{x+1,\ |x|+1\} \). Then which of the following is true?

If $f(x) = \begin{cases} x\tan^{-1}x\cdot\sec^{-1}\!\left(\dfrac{1}{x}\right), & x\in(-1,1)\setminus\{0\} \\ \dfrac{\pi}{2}, & x=0 \end{cases}$, then $f'(0)$ is:

\(\lim_{x \to 4} \frac{\cos x - \cos a}{\cos x - \cot a} = \sin^3 a\)State whether this is true or false.

Evaluate: \[\lim_{x \to 3} \frac{\sqrt{3x} - 3}{\sqrt{2x - 4} - \sqrt{2}}\]

If \(f(c) = 2\), \(f'(c) = 1\), \(g(c) = -1\), \(g'(c) = 2\) then \(\lim_{x \to c} \frac{g(x)f(c) - g(c)f(x)}{x - c} =\) ______

The derivative of \(\tan^{-1}\left(\dfrac{\sin x - \cos x}{\sin x + \cos x}\right)\), with respect to \(\dfrac{x}{2}\), where \(x \in \left(0, \dfrac{\pi}{2}\right)\) is:

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 $f(x) = \begin{cases} \frac{e^{\sin x} - 1}{\sin x} & 0 < x < \frac{\pi}{6} \\ \lambda & x = 0 \end{cases}$ is continuous at $x = 0$, the value of $\frac{\ln(3)}{b^2}$ is equal to

Let \(y = \log \sin(x^2)\), \(0

Let \(f(x)\) be a continuous and differentiable function such that \(\displaystyle\lim_{h\to 0}\frac{f(3+7h)-f(3+4h)}{h} = 4\). Then the value of \(f'(3)\) equals:

Let \(f(x) = [x]\cos\left(\dfrac{2x-1}{2}\right)\pi\), where \([\cdot]\) denotes the greatest integer function. Then \(f\) is:

\(\lim_{h \to 0} \left[\frac{1}{h\sqrt[3]{8+h}} - \frac{1}{2h}\right]\) is equal to

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

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