Evaluate the Limit

Problem

(lim_x→1)(1/(x−1)+1/(x2−3*x+2))

Solution

1. Factor the denominator of the second fraction to find a common denominator.

x2−3*x+2=(x−1)*(x−2)

2. Rewrite the expression using the common denominator (x−1)*(x−2)

1/(x−1)+1/((x−1)*(x−2))

3. Combine the fractions by multiplying the first term by (x−2)/(x−2)

(x−2)/((x−1)*(x−2))+1/((x−1)*(x−2))

4. Simplify the numerator by adding the terms together.

(x−2+1)/((x−1)*(x−2))=(x−1)/((x−1)*(x−2))

5. Cancel the common factor (x−1) from the numerator and the denominator, noting that x≠1 as we approach the limit.

1/(x−2)

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

1/(1−2)=−1

Final Answer

(lim_x→1)(1/(x−1)+1/(x2−3*x+2))=−1

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