Evaluate the Limit

Problem

(lim_x→−5)((x^3+125)/(x+5))

Solution

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

2. Recognize that the numerator x^3+125 is a sum of cubes, which follows the formula a^3+b^3=(a+b)*(a^2−a*b+b^2)

3. Factor the numerator using a=x and b=5

x^3+125=(x+5)*(x^2−5*x+25)

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

(lim_x→−5)(((x+5)*(x^2−5*x+25))/(x+5))

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

(lim_x→−5)(x^2−5*x+25)

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

(−5)^2−5*(−5)+25=25+25+25

7. Calculate the final sum.

25+25+25=75

Final Answer

(lim_x→−5)((x^3+125)/(x+5))=75

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