Differentiation Question 110
Question: $ \underset{x\to 0}{\mathop{\lim }}[ \frac{\sin (sgn(x))}{(sgn(x))} ], $ where [.] denotes the greatest integer function, is equal to
Options:
A) 0
B) 1
C) -1
D) Does not exist
Show Answer
Answer:
Correct Answer: A
Solution:
[a] $ =\underset{x\to 0+}{\mathop{\lim }}[ \frac{\sin x}{ (x)} ]=\underset{x\to 0+}{\mathop{\lim }}[ \frac{\sin 1}{1} ]=0 $
$ =\underset{x\to {0^{-}}}{\mathop{\lim }}[ \frac{\sin (x)}{ (x)} ]=\underset{x \to {0^{-}}}{\mathop{\lim }}[ \frac{\sin (-1)}{-1} ] $
$ =\underset{x\to {0^{-}}}{\mathop{\lim }}[sin1] $ Hence, the given limit is 0.
BETA
