Question

Let $[x^{r}]$ denotes the greatest integer of $x^{r}$ and $|x|$ denotes the modulus of x. Then $\lim_{x\rightarrow0}\frac{\sum_{r=1}^{100}[x^{r}]}{1+|x|}$ is ________.

• A. does not exist
• B. is -1
• C. is 1
• D. is 100

Hint:

For limits involving Greatest Integer Function $[x]$, always check LHL and RHL separately.

Answer 1

The Correct Option is A

Step 1: Concept
Evaluate Left-Hand Limit (LHL) and Right-Hand Limit (RHL).
Step 2: Analysis
As $x \rightarrow 0^+$: $x^r$ is a small positive decimal. Thus, $[x^r] = 0$ for all $r$. Numerator $= 0$. RHL $= 0$.
As $x \rightarrow 0^-$: For odd $r$, $x^r$ is a small negative decimal (e.g., -0.1). Thus, $[x^r] = -1$.
Step 3: Calculation
For odd $r$ (50 values), $[x^r] = -1$. For even $r$ (50 values), $x^r>0$, so $[x^r] = 0$.
Numerator $= 50(-1) + 50(0) = -50$.
LHL $= -50 / (1+0) = -50$.
Step 4: Conclusion
Since RHL ($0$) $\neq$ LHL ($-50$), the limit does not exist.
Final Answer:(A)