List of top Mathematics Questions on Limit and Continuity asked in KEAM

The value of $\lim_{x\rightarrow 2^{+}}\frac{[x]-2}{x-2}$ is

Let $f(x)=\begin{cases}ax+3 & x<1\\ \frac{4x}{a} & x\ge 1\end{cases}$. If $\lim_{x\rightarrow 1}f(x)$ exists, then the possible values of $a$ are

Let $f(x)=-\sqrt{49-x^{2}}$. Then $\lim_{x\rightarrow1}\frac{f(x)-f(1)}{x-1}$ is equal to

Let $[x]$ denote the greatest integer less than or equal to x. Then $\lim_{x\rightarrow0^{-}}\frac{\sin[x]}{[x]}$ is equal to

If the function $f(x)=\begin{cases}\frac{x^{2}}{2},& 0\le x<1\\ 2x^{2}-ax+1.5,& 1\le x\end{cases}$ is continuous on $[0, 5]$, then the value of a is

The value of $\lim_{x\rightarrow\frac{\pi^{-}}{2}}(\tan x-\sec x)$ is equal to

If \([x]\) denotes the greatest integer less than or equal to \(x\), then \[ \lim_{x\to 0^-}\frac{\sin([x])}{[x]} \] is equal to ________.

Let \([a]\) denote the greatest integer less than or equal to \(a\). Then \[ \lim_{x\to 0^{+}} x\left( \left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] \right) \] is equal to ________.

Evaluate the limit: \[ \lim\limits_{x \to 6} \frac{\sqrt{x^2 + 13} - 7}{x^2 - 36}. \]

If \[ \lim_{x \to 9} f(x) = 6 \quad {and} \quad \lim_{x \to 9} g(x) = 3, \] then \[ \lim_{x \to 9} \frac{f(x) - 2g(x)}{g(x)} \] is equal to

Evaluate the limit: $$ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) $$ is equal to 

Let \(f:R→R\) be a function defined by
\(f(x) =   \begin{cases}     3e^x       & \quad \text {if}\  {x<0}\\     x^2+3x+3  & \quad \text {if}\  0≤x<1   \\ x^2-3x-3 & \quad \text {if} \ x≥1\end{cases}\)

\(\lim_{x\to0} e^{x}-\dfrac{1}{3}×(1-e^{2x})\)\(=\)?