Evaluate the Limit

Problem

(lim_x→−1)((x2+6*x+5)/(x2−3*x−4))

Solution

1. Substitute the value x=−1 into the expression to check for an indeterminate form.

((−1)2+6*(−1)+5)/((−1)2−3*(−1)−4)=(1−6+5)/(1+3−4)=0/0

2. Factor the quadratic expression in the numerator by finding two numbers that multiply to 5 and add to 6

x2+6*x+5=(x+1)*(x+5)

3. Factor the quadratic expression in the denominator by finding two numbers that multiply to −4 and add to −3

x2−3*x−4=(x+1)*(x−4)

4. Simplify the rational expression by canceling the common factor (x+1) from the numerator and the denominator.

((x+1)*(x+5))/((x+1)*(x−4))=(x+5)/(x−4)

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

(−1+5)/(−1−4)=4/(−5)

Final Answer

(lim_x→−1)((x2+6*x+5)/(x2−3*x−4))=−4/5

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