Evaluate the Limit limit as x approaches 0 of x/(x^2-x)

Problem

(lim_x→0)(x/(x2−x))

Solution

1. Factor the denominator by taking out the greatest common factor, which is x

x/(x2−x)=x/(x*(x−1))

2. Simplify the expression by canceling the common factor x from the numerator and the denominator, noting that x≠0 as we approach the limit.

x/(x*(x−1))=1/(x−1)

3. Substitute the value x=0 into the simplified expression to evaluate the limit.

(lim_x→0)(1/(x−1))=1/(0−1)

4. Calculate the final numerical value.

1/(−1)=−1

Final Answer

(lim_x→0)(x/(x2−x))=−1

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