Question

Question #3bf5e

Answer 1

Use L'Hôpital's rule

Explanation:

Here is the definition of `|x - 1|`:

`|x - 1| = x - 1; x >= 1`
`|x - 1| = -x + 1; x < 1`

Because the limit is approaching from the positive side only, we can replace `|x - 1|` with (x - 1):

`lim_(xto1^+) (x^2 -1)/(x - 1)`

Because the expression evaluated at the limit results in the indeterminate form `0/0`, the use of L'Hôpital's rule is warrented.

Take the derivative of the numerator:

`(d(x^2 - 1))/dx = 2x`

Take the derivative of the denominator:

`(d(x - 1))/dx = 1`

The rule says that:

`lim_(xto1^+) (2x)/1`

Goes to the same limit as the original expression.

`lim_(xto1^+) 2x = 2`

Therefore, the original limit:

`lim_(xto1^+) (x^2 -1)/|x - 1| = 2`