Question

If $f\left(x\right)=\begin{array}{cc}\frac{\sin \left[x\right]}{\left[x\right]},& \left[x\right]\ne 0\\ 0& \left[x\right]=0\end{array}$ where $[.]$ denotes the greatest integer function.,then $\underset{x\to 0}{lim}f\left(x\right)$ is equal to

• A. $1$
• B. $0$
• C. $-1$
• D. None of these

Answer 1

Answer: (D)

None of these

$f\left(x\right)=\frac{sin\left[x\right]}{\left[x\right]},\left[x\right]\ne 0$

$=0$ , $\left[x\right]=0$

$\underset{x\to 0}{lim}\frac{sin\left[x\right]}{\left[x\right]}=$

Left hand Limit

$\underset{x\to 0}{lim}\frac{sin\left[x\right]}{\left[x\right]}$

$=\underset{h\to 0}{lim}\frac{sin[0-h]}{[0-h]}=\underset{h\to 0}{lim}\frac{sin[-h]}{[-h]}$

$=\underset{h\to 0}{lim}\frac{sin(-1)}{(-1)}$

$=sin1$

Right hand limit

$\underset{x\to 0^{+}}{lim}\frac{sin\left[x\right]}{\left[x\right]}=\underset{h\to 0}{lim}\frac{sin[0+h]}{[0+h]}$

$=\underset{h\to 0}{lim}\frac{sin\left[h\right]}{\left[h\right]}=0$.

$\therefore LHL\ne R.H.L$

$\therefore \underset{x\to 0}{lim}\frac{sin\left[x\right]}{\left[x\right]}$ does n't exist.