Question

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

Answer 1

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`