Evaluate the Limit

Problem

(lim_x→−4)((x3+64)/(x+4))

Solution

1. Identify the indeterminate form by substituting x=−4 into the expression, which results in ((−4)3+64)/(−4+4)=0/0

2. Factor the numerator using the sum of cubes formula, a3+b3=(a+b)*(a2−a*b+b2) where a=x and b=4

x3+64=(x+4)*(x2−4*x+16)

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

((x+4)*(x2−4*x+16))/(x+4)=x2−4*x+16

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

(−4)2−4*(−4)+16=16+16+16

5. Calculate the final sum.

16+16+16=48

Final Answer

(lim_x→−4)((x3+64)/(x+4))=48

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