Practicing Success
If $f(x)=\left\{\begin{array}{r}x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+...+\left[\frac{n}{x}\right]\right), &x ≠ 0 \\ k \quad \quad ~~~~~~~~~~, & x=0\end{array}\right.$ and $n \in N$. Then the value of k for which f(x) is continuous at x = 0, is |
$n$ $n+1$ $n(n+1)$ $\frac{n(n+1)}{2}$ |
$\frac{n(n+1)}{2}$ |
For any x > 0, we know that $\frac{1}{x}-1<\left[\frac{1}{x}\right] \leq \frac{1}{x} \Rightarrow 1-x<x\left[\frac{1}{x}\right] \leq 1$ $\frac{2}{x}-1<\left[\frac{2}{x}\right] \leq \frac{2}{x} \Rightarrow 2-x<x\left[\frac{2}{x}\right] \leq 2$ $\vdots \quad \vdots \quad \vdots \quad \vdots \quad \vdots$ $\frac{n}{x}-1<\left[\frac{n}{x}\right] \leq \frac{n}{x} \Rightarrow n-x<x\left[\frac{n}{x}\right] \leq n$ ∴ $\frac{n(n+1)}{2}-n x<x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+...+\left[\frac{n}{x}\right]\right) \leq \frac{n(n+1)}{2}$ But, $\lim\limits_{x \rightarrow 0}\left\{\frac{n(n+1)}{2}-n x\right\}=\frac{n(n+1)}{2}$ and, $\lim\limits_{x \rightarrow 0} \frac{n(n+1)}{2}=\frac{n(n+1)}{2}$ Therefore, by Sandwhich theorem, we obtain $\lim\limits_{x \rightarrow 0} x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+...+\left[\frac{n}{x}\right]\right)=\frac{n(n+1)}{2}$ |