Evaluate the Limit

Problem

(lim_x→2)((x4−16)/(x−2))

Solution

1. Identify the indeterminate form by substituting x=2 into the expression, which results in (2−16)/(2−2)=0/0

2. Factor the numerator using the difference of squares formula a2−b2=(a−b)*(a+b)

x4−16=(x2−4)*(x2+4)

3. Factor the term (x2−4) further using the difference of squares formula again.

(x2−4)*(x2+4)=(x−2)*(x+2)*(x2+4)

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

(lim_x→2)(((x−2)*(x+2)*(x2+4))/(x−2))

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

(lim_x→2)(x+2)*(x2+4)

6. Evaluate the limit by substituting x=2 into the remaining expression.

(2+2)*(2+4)=(4)*(8)=32

Final Answer

(lim_x→2)((x4−16)/(x−2))=32

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