Limits, Continuity & Differentiability
Limit involving floor/greatest integer function
nta_pyq_2023_jan
Grade 12
Question:
The set of all values of a for which \lim_{x \to a}\left([x-5] - [2x+2]\right) = 0, where [\alpha] denotes the greatest integer less than or equal to \alpha, is equal to
(-7.5, -6.5)
(-7.5, -6.5]
[-7.5, -6.5]
[-7.5, -6.5)
Step-by-Step Solution
Key Concept: For the limit to exist, [x-5]-[2x+2] must be constant near x = a. Simplify to [x]-[2x] = 7 and analyze integer and non-integer a separately.
Condition reduces to [a]-[2a]=7. For integer a: a=-7 works. For non-integer a = I+f: cases give a \in (-7,-6.5) \cup \{-7\} \cup (-7.5,-7) = (-7.5,-6.5).
Correct Answer: 1