Functions Question 405
Question: If [.] denotes the greatest integer less than or equal to x, then the value of $ \underset{x\to 1}{\mathop{\lim }},(1-x+[x-1]+[1-x]) $ is
Options:
A) 0
B) 1
C) -1
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
We have $ \underset{x\to 1-}{\mathop{\lim }},(1-x+[x-1]+[1-x]) $
$ =\underset{h\to 0}{\mathop{\lim }},(1-(1-h)+[1-h-1]+[1-(1-h)]), $
$ =\underset{h\to 0}{\mathop{\lim }}(h+[-h]+[h])=\underset{h\to 0}{\mathop{\lim }}(h-1+0)=-1 $
and $ \underset{x\to 1+}{\mathop{\lim }}(1-x+[x-1]+[1-x]), $
$ =\underset{h\to 0}{\mathop{\lim }}(1-(1+h)+[1+h-1]+[1-(1+h)]) $
$ =\underset{h\to 0}{\mathop{\lim }}(-h+[h]+[-h])=\underset{h\to 0}{\mathop{\lim }}(-h+0-1)=-1 $
\ $ \underset{x\to 1}{\mathop{\lim }}f(x)=-1 $ .
BETA
