Evaluate the Limit

Problem

(lim_x→1)(1/(1−x)−2/(1−x2))

Solution

1. Factor the denominator of the second fraction using the difference of squares formula 1−x2=(1−x)*(1+x)

1/(1−x)−2/((1−x)*(1+x))

2. Find a common denominator by multiplying the first fraction by (1+x)/(1+x)

(1+x)/((1−x)*(1+x))−2/((1−x)*(1+x))

3. Combine the fractions over the common denominator and simplify the numerator.

(1+x−2)/((1−x)*(1+x))

(x−1)/((1−x)*(1+x))

4. Simplify the expression by factoring out a negative sign from the numerator to cancel the common factor (1−x)

(−(1−x))/((1−x)*(1+x))

(−1)/(1+x)

5. Evaluate the limit by substituting x=1 into the simplified expression.

(−1)/(1+1)

−1/2

Final Answer

(lim_x→1)(1/(1−x)−2/(1−x2))=−1/2

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