Limits, Continuity & Differentiability Question - None

Question

For each

x \in \mathbb{R}

, let

$ [x] $

be the greatest integer less than or equal to x. Then

\lim_{x \to 0} \frac{x[|x|] \sin[x]}{|x|}

is equal to

Accepted Answer

Step 1: Analyze the left-hand limit (x → 0⁻)

For small negative x (say -0.5 < x < 0):

• |x| = -x, and for small negative x: [|x|] = [-x] = 0 (since 0 < -x < 1)

• [x] = -1 (since x ∈ [-1, 0))

• sin[x] = sin(-1)

The expression becomes: $\frac{x \cdot 0 \cdot \sin(-1)}{|x|} = \frac{0}{|x|} = 0$

Step 2: Analyze the right-hand limit (x → 0⁺)

For small positive x (say 0 < x < 1):

• |x| = x, and for small positive x: [|x|] = [x] = 0 (since 0 < x < 1)

• [x] = 0 (since x ∈ [0, 1))

• sin[x] = sin(0) = 0

The expression becomes: $\frac{x \cdot 0 \cdot 0}{|x|} = \frac{0}{|x|} = 0$

Step 3: Conclusion

Since both the left-hand limit and right-hand limit equal 0, and the function is 0 for all x in a deleted neighborhood of 0:

$$\lim_{x o 0} \frac{x[|x|] \sin[x]}{|x|} = 0$$

∴ Answer: 2

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