List of top Mathematics Questions on Limits asked in TS EAMCET

$[y]$ represents the greatest integer less than or equal to $y$ and $\{y\}$ represents the fractional part of $y$. If \[ \lim_{x \to 0^{+}} \left( [1-x] + \frac{a^{2[1-x] + \{1-x\} - 1}}{2[1-x] + \{1-x\}^{2}} \right) = 11, \] then $a =$ ?

If \[ f(x)= \frac{\lambda e^{\frac1x}+3e^{-\frac1x}} {(\lambda+2)e^{\frac1x}-e^{-\frac1x}}, \qquad x\neq0 \] and $f(0)=k$, $k\in\mathbb R$, is a continuous function at $x=0$, then $2\lambda=$ ?

If a function $ f : (-\infty, 2) \to \mathbb{R} $ is defined by \[ f(x)= \begin{cases} \dfrac{\alpha \left|x^2 - 3x + 2\right|}{x - 1}, & x < 1 \\[8pt] \dfrac{\sin([x] - x)}{x - [x]}, & x > 1 \\[8pt] \beta, & x = 1 \end{cases} \] and $ f $ is continuous at $ x = 1 $, then find \[ \frac{\alpha^2 + \beta^2}{|\alpha \beta|}. \]

Evaluate \[ \lim_{x\to 0}\frac{\log(4+x)^x-\log 4^x}{\sin^2 x}. \]

Let \[ f(x)=\frac{([x]+|x|-x)x}{\sin|x|} \] be a real valued function. If \[ \alpha=\lim_{x\to0^-}f(x),\qquad \beta=\lim_{x\to0^+}f(x) \] then

Let $f:[-1,2] \to \mathbb{R}$ be defined by $f(x) = [x^2-3]$ where $[.]$ denotes greatest integer function, then the number of points of discontinuity for the function $f$ in $(-1,2)$ is

$\lim_{x \to 0} \frac{\sqrt[3]{\cos x} - \sqrt{\cos x}}{\sin^2 x} =$

If the real valued function $ f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x \sin x}, & \text{if } x < 0 \\ p, & \text{if } x = 0 \\ \frac{\log(1 + q \sin x)}{x}, & \text{if } x > 0 \end{cases} $ is continuous at $ x = 0 $, then $ p + q = $

If $[t]$ represents the greatest integer $\leq t$ then the value of $\lim_{x\to 3} \frac{11-[2-x]}{[x+10]}$ is

For $a\neq0$ and $b\neq0$, if the real valued function $f(x) = \frac{\sqrt[4]{625+4x}-5}{\sqrt[4]{625+5bx}-5}$ is continuous at $x=0$, then $f(0) =$

If $\{x\}=x-[x]$ where $[x]$ is the greatest integer $\le x$ and $\lim_{x\to 0^+} \frac{\text{Cos}^{-1}(1-\{x\}^2)\text{Sin}^{-1}(1-\{x\})}{\{x\}-\{x\}^3} = \theta$, then $\tan\theta=$