Check the continuity of the function $f$ given by $f(x) = 2x + 3$ at $x = 1$.

Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \alpha+\frac{\sin [x]}{x}, & x>0 \\ 2, & x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right], & x < 0 \end{cases}$. If $f$ is continuous at $x=0$,find the value of $\alpha + \beta$.

If the function $f(x)$ defined by $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$,then $k = . . . . . .$

The function $f(x) = \frac{x^2 - 1}{x^3 - 1}$ is not defined at $x = 1$. If the function is continuous at $x = 1$,then the value of $f(1)$ will be:

Let $[x]$ denote the greatest integer less than or equal to $x$ and $f(x) = [\tan^2 x]$. Which of the following is true?

If the function $f(x) = \begin{cases} \frac{x^2-(A+2)x+A}{x-2} & \text{for } x \neq 2 \\ 2 & \text{for } x=2 \end{cases}$ is continuous at $x=2$, then: