Question

Let the function $f(x) = \sin[x]$, where $[\cdot]$ is the greatest integer function and $g(x) = |x|$.

What is $\lim_{x\to 0} \{f(x)g(x)\}$ equal to?

• A. -1
• B. 0
• C. 1
• D. Limit does not exist

Answer 1

Answer: (B)

0

Answer 2

Solution Overview:

We are given:

$$f(x) = \sin([x]) \quad \text{and} \quad g(x)=|x|$$

where $[x]$ is the floor (greatest integer) function.

1. For $x \to 0^+$:

   • $0 \leq x < 1$ implies $[x] = 0$.
   • Hence, $f(x)=\sin(0)=0$, and so $f(x)g(x)=0\cdot |x|=0$.

2. For $x \to 0^-$:

   • $-1<x<0$ implies $[x] = -1$.
   • Hence, $f(x)=\sin(-1)$ (a constant value),
   • But $g(x)=|x| \to 0$.
   • So, $f(x)g(x)=\sin(-1)|x| \to 0$.

Since both one-sided limits equal 0, the overall limit is 0.