Evaluate the Limit

Problem

(lim_x→2)((x5−32)/(x−2))

Solution

1. Identify the form of the limit by substituting x=2 into the expression.

(2−32)/(2−2)=0/0

2. Recognize that the numerator is a difference of powers in the form an−bn where a=x b=2 and n=5

x5−2=(x−2)*(x4+x(2)3+x(2)2+x(2)+2)

3. Simplify the expression by canceling the common factor (x−2) from the numerator and the denominator.

((x−2)*(x4+2*x3+4*x2+8*x+16))/(x−2)=x4+2*x3+4*x2+8*x+16

4. Evaluate the limit by substituting x=2 into the simplified polynomial.

2+2*(2)+4*(2)+8*(2)+16

5. Calculate the final sum of the terms.

16+16+16+16+16=80

Final Answer

(lim_x→2)((x5−32)/(x−2))=80

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