List of top Mathematics Questions on limits and derivatives asked in MHT CET

Let the function $f(x)$ be defined as \[ f(x)=\frac{x-|x|}{x} \] then:
Let the function \[ f(x)=\frac{x-|x|}{x} \] Then
If triangle ABC is a right angled at A and \( \tan \frac{\text{B}}{2}, \tan \frac{\text{C}}{2} \) are roots of the equation \( a x^2 + bx + c = 0, \text{a} \ne 0 \), then

Define \[ f(x) = \begin{cases} b - ax, & \text{if } x < 2 \\ 3, & \text{if } x = 2 \\ a + 2bx, & \text{if } x > 2 \end{cases} \]

If \( \lim_{x \to 2} f(x) \) exists, then find \( \frac{a}{b} \).

$\lim_{x \to 0} \frac{e^{\tan x} - e^x}{\tan x - x} =$}

Define \[ f(x) = \begin{cases} b - ax, & \text{if } x < 2 \\ 3, & \text{if } x = 2 \\ a + 2bx, & \text{if } x > 2 \end{cases} \]

If \( \lim_{x \to 2} f(x) \) exists, then find \( \frac{a}{b} \).

If triangle ABC is a right angled at A and \( \tan \frac{\text{B}}{2}, \tan \frac{\text{C}}{2} \) are roots of the equation \( a x^2 + bx + c = 0, \text{a} \ne 0 \), then

Define \[ f(x) = \begin{cases} b - ax, & \text{if } x < 2 \\ 3, & \text{if } x = 2 \\ a + 2bx, & \text{if } x > 2 \end{cases} \]

If \( \lim_{x \to 2} f(x) \) exists, then find \( \frac{a}{b} \).

Let $f(x) = 5 - |x-2|$ and $g(x) = |x + 1|$, $x \in \mathbb{R}$. If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta$, then $\lim_{x \to \alpha\beta} \frac{(x-1)(x^2-5x+6)}{x^2-6x+8}$ is equal to}
If $f(a) = 2$, $f'(a) = 1$, $g(a) = -1$, $g'(a) = 2$, then as $x$ approaches $a$, $\frac{g(x)f(a)-g(a)f(x)}{(x-a)}$ approaches}
If $y = \left[(x+1)(2x+1)(3x+1)\cdots (nx+1)\right]^n$, then $\left.\frac{dy}{dx}\right|_{x=0}$ is