List of top Mathematics Questions on limits and derivatives

If $ f(x) = x^2 $, find $ \frac{f(1.1) - f(1)}{1.1 - 1} $.

Let $\left\lfloor t \right\rfloor$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbb{N}$ for which

\[ \lim_{x \to 0^+} \left( x \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \dots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \dots + \left\lfloor \frac{9^2}{x^2} \right\rfloor \right) \geq 1 \]

is equal to __________.

Find the derivative of \[ \frac{d}{dx} \left( \lim_{n \to 1} \frac{x^n - 1}{n+1} \right). \]

Find the derivative: \[ \frac{d}{dx} \left[ \left| \frac{x}{4} \right| \right] \]

Find the derivative of $ \log(x^9) $.

$\frac{d}{dx} \left[ \log x^2 + \log a^2 \right] $

Let \[ f(x) = \int_{0}^{x} \left( t + \sin\left(1 - e^t\right) \right) \, dt, \, x \in \mathbb{R}. \] Then \[ \lim_{x \to 0} \frac{f(x)}{x^3} \] is equal to:

The integral $\int_{0}^{\pi/4} \frac{136 \sin x}{3 \sin x + 5 \cos x} \, dx$ is equal to

$\lim_{x \to 0} \frac{e - (1 + 2x)^{\frac{1}{2x}}}{x} \quad \text{is equal to:}$

Let \[ \int_{\log_e a}^{4} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6}. \] Then $e^\alpha$ and $e^{-\alpha}$ are the roots of the equation:

Let $I(x) = \int \frac{6}{\sin^2 x (1 - \cot x)^2} dx$. If $I(0) = 3$, then $I\left(\frac{\pi}{12}\right)$ is equal to:

If $ \lim_{x \to 4} \frac{2x^2 + (3+2a)x + 3a{x^3 - 2x^2 - 23x + 60} = \frac{11}{9} $, then find $ \lim_{x \to a} \frac{x^2 + 9x + 20}{x^2 - x - 20} $.}

If the function $ f(x) $ is given by \[ f(x) = \begin{cases} \frac{\tan(a(x-1))}{\frac{x-1}{x}}, & tif0<x<1
\frac{x^3-125}{x^2 - 25} , & \text{if } 1 \leq x \leq 4
\frac{b^x - 1}{x}, & \text{if } x>4 \end{cases} \] is continuous in its domain, then find $ 6a + 9b^4 $.

Evaluate the following limit: $ \lim_{n \to \infty} \prod_{r=3}^n \frac{r^3 - 8}{r^3 + 8} $.

If the value of the integral \[ \int_{-1}^{1} \frac{\cos \alpha x}{1 + 3^x} \, dx = \frac{2}{\pi}, \] then a value of $ \alpha $ is:

The value of $\int_{-\pi}^{\pi} \frac{2y(1 + \sin y)}{1 + \cos^2 y} \, dy$

Let $ P = \{ z \in \mathbb{C} : |z + 2 - 3i| \leq 1 \} $ and $ Q = \{ z \in \mathbb{C} : z(1 + i) + \overline{z}(1 - i) \leq -8 \} $.
Let $ z $ in $ P \cap Q $ have $ |z - 3 + 2i| $ be maximum and minimum at $ z_1 $ and $ z_2 $, respectively.
If $ |z_1|^2 + 2|z_2|^2 = \alpha + \beta \sqrt{2} $, where $ \alpha $ and $ \beta $ are integers, then $ \alpha + \beta $ equals ____.

Let $\{x\}$ denote the fractional part of $x$, and $f(x) = \frac{\cos^{-1}(1 - \{x\}^2) \sin^{-1}(1 - \{x\})}{\{x\} - \{x\}^3}, \quad x \neq 0$.If $L$ and $R$ respectively denote the left-hand limit and the right-hand limit of $f(x)$ at $x = 0$, then $\frac{32}{\pi^2} \left(L^2 + R^2\right)$ is equal to $\_\_\_\_\_\_\_\_$.

If $ f(x) $ is defined as follows:
$$ f(x) = \begin{cases} 4, & \text{if } -\infty < x < -\sqrt{5}, \\ x^2 - 1, & \text{if } -\sqrt{5} \leq x \leq \sqrt{5}, \\ 4, & \text{if } \sqrt{5} \leq x < \infty. \end{cases} $$ If $ k $ is the number of points where $ f(x) $ is not differentiable, then $ k - 2 = $

Let $ f(x) = x^2 \log(\cos x) \log(1 + x) $ for $ x \neq 0 $, and $ f(0) = 0 $. Determine the behavior of $ f(x) $ at $ x = 0 $.

Let $ f $ be the function defined by:
\[ f(x) = \begin{cases} \frac{x^2 - 1}{x^2 - 2|x-1| - 1}, & \text{if } x \neq 1, \\ \frac{1}{2}, & \text{if } x = 1. \end{cases} \] The function is continuous at:

Let the function $ g: (-\infty, 0) \rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $ be given by $ g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2} $. Determine the properties of $ g $.

If α > β > 0 are the roots of the equation ax2 + bx + 1 = 0, and $\lim\limits_{x \to \frac{1}{\alpha}}\left(\frac{1-cos(x^2+bx+a)}{2(1-ax)^2}\right)^{1/2}=\frac{1}{k}\left(\frac{1}{\beta}-\frac{1}{\alpha}\right),$ then k is equal to

The derivative $\frac{dy}{dx} $ of the function y = sin(cos(log(x))) is,

$ \lim_{x \to 0} \frac{x \cdot \cot(4x)}{\sin(2x) \cdot \cot^2(2x)} = ? $