What is #lim_(xrarr-1) (1/x-1)/(x-1)#?

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Answer 1 · 2015-11-02T02:16:34

#lim_(xrarr-1) (1/x-1)/(x-1)=1#

#lim_(xrarr-1) (1/x-1)/(x-1) = (lim_(xrarr-1) (1/x-1))/(lim_(xrarr-1)(x-1))# Provided that both limits exist and the denominator doses not go to #0#.

So, consider the limits separately.

#lim_(xrarr-1) (1/x-1) = lim_(xrarr-1) 1/x-lim_(xrarr-1) 1#

# = 1/-1-1=-1-1=-2#

So the numerator limit exists and equals #-2#

#lim_(xrarr-1)(x-1) = -1-1=-2#

So the denominator limit exists and equals #-2#

Therefore

#lim_(xrarr-1) (1/x-1)/(x-1)= (-2)/(-2) =1#