Evaluate the Limit

Problem

(lim_x→8)((x2−3*x−40)/(x−8))

Solution

1. Identify the form of the limit by substituting x=8 into the expression.

(8−3*(8)−40)/(8−8)=(64−24−40)/0=0/0

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

2. Factor the quadratic trinomial in the numerator by finding two numbers that multiply to −40 and add to −3

x2−3*x−40=(x−8)*(x+5)

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

(lim_x→8)(((x−8)*(x+5))/(x−8))

4. Simplify the expression by canceling the common factor (x−8) from the numerator and the denominator.

(lim_x→8)(x+5)

5. Evaluate the limit by substituting x=8 into the simplified expression.

8+5=13

Final Answer

(lim_x→8)((x2−3*x−40)/(x−8))=13

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