How do you find the limit of #abs(x-4) / (x-4)# x as it approaches 4 from the left and right?

1 Answer
Sep 26, 2015

See the explanation.

Explanation:

Let #epsilon in R^+, epsilon->0#

When #x->4+epsilon# then #x-4>0# and hence #|x-4|=x-4#
When #x->4-epsilon# then #x-4<0# and hence #|x-4|=-x+4#

Form the right:
#lim_(x->4+epsilon) |x-4|/(x-4) = lim_(x->4+epsilon) (x-4)/(x-4)=1#

From the left:
#lim_(x->4-epsilon) |x-4|/(x-4) = lim_(x->4-epsilon) (-x+4)/(x-4)=#

#=lim_(x->4-epsilon) (-(x-4)/(x-4))=-1#