Evaluate the Limit

Problem

(lim_x→4)((x^3−64)/(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. Recognize the numerator as a difference of cubes, x^3−4^3 which follows the formula a^3−b^3=(a−b)*(a^2+a*b+b^2)

3. Factor the numerator using the difference of cubes formula where a=x and b=4

(x^3−64)/(x−4)=((x−4)*(x^2+4*x+16))/(x−4)

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

((x−4)*(x^2+4*x+16))/(x−4)=x^2+4*x+16

5. Evaluate the limit by substituting x=4 into the simplified polynomial expression.

(lim_x→4)(x^2+4*x+16)=4^2+4*(4)+16

6. Calculate the final numerical value.

16+16+16=48

Final Answer

(lim_x→4)((x^3−64)/(x−4))=48

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