Evaluate the Limit

Problem

(lim_x→5)((x2+3*x−40)/(x2−25))

Solution

1. Identify the indeterminate form by substituting x=5 into the expression, which results in (5+3*(5)−40)/(5−25)=0/0

2. Factor the numerator x2+3*x−40 into (x+8)*(x−5) by finding two numbers that multiply to −40 and add to 3

3. Factor the denominator x2−25 as a difference of squares into (x+5)*(x−5)

4. Simplify the expression by canceling the common factor (x−5) from the numerator and the denominator.

5. Evaluate the limit by substituting x=5 into the remaining simplified expression (x+8)/(x+5)

Final Answer

(lim_x→5)((x2+3*x−40)/(x2−25))=13/10

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