Evaluate the Limit

Problem

(lim_x→−1)((2*x2−x−3)/(x+1))

Solution

1. Check for direct substitution by plugging x=−1 into the expression.

(2*(−1)2−(−1)−3)/(−1+1)=(2+1−3)/0=0/0

Since this results in an indeterminate form, the expression must be simplified.

2. Factor the numerator 2*x2−x−3 by looking for two numbers that multiply to −6 and add to −1

2*x2−3*x+2*x−3

x*(2*x−3)+1*(2*x−3)

(2*x−3)*(x+1)

3. Substitute the factored form back into the limit expression.

(lim_x→−1)(((2*x−3)*(x+1))/(x+1))

4. Cancel the common factor (x+1) from the numerator and the denominator.

(lim_x→−1)(2*x−3)

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

2*(−1)−3=−5

Final Answer

(lim_x→−1)((2*x2−x−3)/(x+1))=−5

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