Evaluate the Limit

Problem

(lim_x→5)((x3−125)/(x−5))

Solution

1. Identify the indeterminate form by substituting x=5 into the expression, which results in (5−125)/(5−5)=0/0

2. Factor the numerator using the difference of cubes formula, a3−b3=(a−b)*(a2+a*b+b2) where a=x and b=5

x3−125=(x−5)*(x2+5*x+25)

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

((x−5)*(x2+5*x+25))/(x−5)=x2+5*x+25

4. Evaluate the limit by substituting x=5 into the remaining quadratic expression.

5+5*(5)+25=25+25+25

25+25+25=75

Final Answer

(lim_x→5)((x3−125)/(x−5))=75

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