JEE Main 2020 · 3 September · morning · Single correct · +4 / −1

Let [t] denote the greatest integer ≤ t. If for some
λ ∈ R - 1, 0, lim _(x → 0) | (1 - x + | x |)/(λ - x + [ x ]) | = L, then L is equal to :

  1. (a) 1
  2. (b) 2
  3. (c) 0
  4. (d) 1/2
Answer and solution

Answer (b) 2

Solution

Maths sir · 4 steps

The floor function jumps at 0, so match the left and right limits; that fixes λ, and then L follows.

  1. Right-hand limit. For small x > 0: |x| = x and [x] = 0. The numerator is 1 - x + x = 1 and the denominator is λ - x → λ, so the limit is 1/|λ|.
  2. Left-hand limit. For small x < 0: |x| = -x and [x] = -1. The numerator is 1 - 2x → 1 and the denominator is λ - x - 1 → λ - 1, so the limit is 1/(|λ - 1|).

    1/|λ| = 1/(|λ - 1|) ⇒ |λ| = |λ - 1| ⇒ λ = 1/2

  3. Why they must match. L exists only if the two one-sided limits are equal. λ must be the same distance from 0 and from 1, so λ = 1/2. This value is allowed, since λ ≠ 0, 1.
  4. So. L = 1/(|1/2|) = 2, which is option (b).

The idea to keep. When [x] or |x| sits inside a limit at an integer, always split into left and right limits. Demanding that they match is often what finds the unknown constant.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Limits

Asked about 110 times in JEE Main since 2020. This question is from 2020.