Differentiation Question 169
Question: $ \underset{x\to 0}{\mathop{\lim }}[ \frac{\sin [x-3]}{[x-3]} ], $ where $ [.] $ denotes greatest integer function is
Options:
A) 0
B) 1
C) Does not exist
D) sin 1
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ \underset{x\to 0}{\mathop{\lim }}[ \frac{\sin [x-3]}{[x-3]} ] $ For $ x\to {0^{+}},[x-3]=-3 $
$ \therefore \frac{\sin [x-3]}{[x-3]}=\frac{\sin (-3)}{-3}=\frac{\sin 3}{3}\in (0,1) $
$ \therefore \underset{x\to {0^{+}}}{\mathop{\lim }}\frac{\sin [x-3]}{[x-3]}=0 $ For $ x\to {0^{-}},[x-3]=-4 $
$ \therefore \frac{\sin [x-3]}{[x-3]}=\frac{\sin 4}{4} $ lies in (-1, 0)
$ \therefore \underset{x\to {0^{-}}}{\mathop{\lim }}[ \frac{\sin [x-3]}{[x-3]} ]=-1 $
$ \therefore $ Limit does not exist.
BETA
