Evaluate the Limit

Problem

(lim_x→5)((x2−25)/(x2+x−30))

Solution

1. Substitute the value x=5 into the expression to check for an indeterminate form.

(5−25)/(5+5−30)=0/0

2. Factor the numerator using the difference of squares formula a2−b2=(a−b)*(a+b)

x2−25=(x−5)*(x+5)

3. Factor the denominator by finding two numbers that multiply to −30 and add to 1

x2+x−30=(x−5)*(x+6)

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

((x−5)*(x+5))/((x−5)*(x+6))=(x+5)/(x+6)

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

(5+5)/(5+6)=10/11

Final Answer

(lim_x→5)((x2−25)/(x2+x−30))=10/11

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