Evaluate the Limit

Problem

(lim_x→9)((x2−9*x)/(x2−8*x−9))

Solution

1. Substitute the value x=9 into the expression to check for an indeterminate form.

(9−9*(9))/(9−8*(9)−9)=(81−81)/(81−72−9)=0/0

2. Factor the numerator by taking out the greatest common factor x

x2−9*x=x*(x−9)

3. Factor the quadratic trinomial in the denominator by finding two numbers that multiply to −9 and add to −8

x2−8*x−9=(x−9)*(x+1)

4. Simplify the expression by canceling the common factor (x−9) from the numerator and the denominator, noting that x≠9 as we approach the limit.

(x*(x−9))/((x−9)*(x+1))=x/(x+1)

5. Evaluate the limit by substituting x=9 into the simplified expression.

(lim_x→9)(x/(x+1))=9/(9+1)

6. Calculate the final numerical value.

9/10=0.9

Final Answer

(lim_x→9)((x2−9*x)/(x2−8*x−9))=9/10

Want more problems? Check here!