List of top Mathematics Questions on Limit and Continuity

If $\lim_{n\rightarrow5}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=k$, then $\lim_{n\rightarrow k^{+}}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=$

$\lim_{x\rightarrow1}\frac{(9x-1)(\sqrt{x}-1)}{3x^{2}+2x-5}=$

\[ \lim_{x\to 0}\left(\frac{1}{x}-\frac{1}{\sin x}+e^{\frac{1-\cos x}{x}}\right) = \underline{} \] rounded off to one decimal place.

If $\lim_{x \to 3} \left( \frac{x^2 - ax - 3a}{x - 3} \right) = 5$, then $a + b =$

If $f(x) = \begin{cases} x^2 - 1 & \text{if } x \ge 2 \\ x + 1 & \text{if } x<2 \end{cases}$, then $\lim_{x \to 2^+} f(x) + \lim_{x \to 2^-} f(x) = $

If $f(x) = \begin{cases} ax + 7 & \text{if } x<1 \\ 2x - 3 & \text{if } x = 1 \\ \frac{x+b}{b} & \text{if } x>1 \end{cases}$ is continuous at $x = 1$, then

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

Find the value of \(k\) if the function \(f(x)=\dfrac{k\sin x}{x}\) for \(x\neq0\) and \(f(0)=3\) is continuous at \(x=0\).

If $f(x) = 3x - b$, $x>1$ ; $f(x) = 11$, $x = 1$ ; $f(x) = -3x - 2b$, $x<1$ is continuous at $x = 1$, then the values of $a$ and $b$ are :

If \( f(x) = \begin{cases} \frac{\sin^2 ax}{x^2}, & \text{if } x \neq 0 \\ 1, & \text{if } x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of 'a' is :

Evaluate the limit: \[ L = \lim_{x \to 0} \frac{35^x - 7^x - 5^x + 1}{(e^x - e^{-x}) \ln(1 - 3x)} \]