Question

If \([\,]\) denotes greatest integer function, then \[ \lim_{x\to -\frac{3}{5}}\frac{1}{x}\left[\frac{-1}{x}\right] \] is

• A. \(-\dfrac{5}{3}\)
• B. \(\dfrac{5}{3}\)
• C. \(\dfrac{10}{3}\)
• D. \(-\dfrac{10}{3}\)

Hint:

For greatest integer function problems, first find the nearby constant value of the expression inside the brackets before evaluating the limit.

Answer 1

The Correct Option is A

Step 1: Evaluate the expression inside the greatest integer function.
We have
\[ \frac{-1}{x} \] As
\[ x\to -\frac{3}{5} \] we get
\[ \frac{-1}{x}\to \frac{-1}{-\frac{3}{5}} \] \[ =\frac{5}{3} \] Now,
\[ \frac{5}{3}=1.666\ldots \] Therefore, its greatest integer value is
\[ \left[\frac{-1}{x}\right]=1 \] for all \(x\) sufficiently close to \(-\dfrac{3}{5}\).

Step 2: Simplify the given limit.
Thus, the expression becomes
\[ \lim_{x\to -\frac{3}{5}}\frac{1}{x}(1) \] \[ = \lim_{x\to -\frac{3}{5}}\frac{1}{x} \]

Step 3: Evaluate the limit.
Substituting \(x=-\dfrac{3}{5}\),
\[ \frac{1}{x} = \frac{1}{-\frac{3}{5}} \] \[ = -\frac{5}{3} \]

Step 4: Final conclusion.
Hence,
\[ \boxed{-\frac{5}{3}} \]