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 (4−64)/(4−4)=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=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+4*(4)+16=16+16+16

16+16+16=48

Final Answer

(lim_x→4)((x3−64)/(x−4))=48

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