npenes lim _(xarrow 1)(x^3-x)/(x^2)-2x+1

Question

Answer 1

To evaluate the limit

, we first simplify the expression.Notice that the denominator can be factored:

Next, we factor the numerator:

So the limit becomes:

We can cancel the common factor \((x - 1)\) in the numerator and denominator:

Now, substitute

into the simplified expression:

Since this results in an indeterminate form

, we need to re-evaluate our simplification. Let's instead use L'Hôpital's Rule, which is applicable for limits of the form

.Differentiate the numerator and the denominator:

Now apply L'Hôpital's Rule:

Substitute

into the new expression:

This still results in an indeterminate form. Let's try another approach by factoring and simplifying directly.Factor the numerator:

Factor the denominator:

Cancel the common factor \((x - 1)\):

Now, substitute

:

Since this still results in an indeterminate form, we conclude that the limit does not exist in a finite sense. Therefore, the limit is undefined.

Вопрос