Question

Does the limit liₓ→₂​ 1/⌊x⌋​ exist? Provide justification for your claim. Here ⌊x⌋ denotes the "floor" function, which is the largest integer less than or equal to x.

Answer 1

The limit limₓ→₂ 1/⌊x⌋ does not exist because the limits from the left and right sides are different.

Step-by-step explanation:

The limit limₓ→₂ 1/⌊x⌋ does not exist.

When x approaches 2 from the left side (x → 2⁻), ⌊x⌋ becomes 1, and the expression becomes 1/1 = 1.

However, when x approaches 2 from the right side (x → 2⁺), ⌊x⌋ becomes 2, and the expression becomes 1/2 = 0.5.

Since the limits from the left and right sides are different, the overall limit does not exist.