# Let F(x) = (x^2-1)/|x-1|. Find the limit as x aproaches 1 from the right of F(x) and limit as x approaches 1 from the left of F(x).

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### 2 Answers

The function `f(x) = (x^2 -1)/(|x - 1|)` .

To determine the limit `lim_(x->1) f(x)` from both the directions keep in mind that

|x - 1| = (x - 1) when x > 1

|x - 1| = (1 - x) when x < 1

`lim_(x->1^+)f(x)`

= `lim_(x->1^+)(x^2 -1)/(|x - 1|)`

= `lim_(x->1^+)((x - 1)(x + 1))/(x - 1)`

= `lim_(x->1^+)(x + 1)`

= 2

`lim_(x->1^-)f(x)`

= `lim_(x->1^-)(x^2 -1)/(|x - 1|)`

= `lim_(x->1^-)((x - 1)(x + 1))/(1 - x)`

= `lim_(x->1^-)-(x + 1)`

= -2

The function .

To determine the limit from both the directions keep in mind that

|x - 1| = (x - 1) when x > 1

|x - 1| = (1 - x) when x < 1

=

=

=

= 2

=

=

=

= -2

Ok Great! Makes sense. I was forgeting to factor out. What does this look like on a graph?