Evaluate the Limit

Problem

(lim_x→4)((x2−16)/(4−x))

Solution

1. Identify the indeterminate form by substituting x=4 into the expression, which results in 0/0

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

x2−16=(x−4)*(x+4)

3. Rewrite the denominator to facilitate cancellation by factoring out a negative sign.

4−x=−(x−4)

4. Substitute these expressions back into the limit.

(lim_x→4)(((x−4)*(x+4))/(−(x−4)))

5. Simplify the expression by canceling the common factor (x−4) assuming x≠4

(lim_x→4)((x+4)/(−1))

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

(4+4)/(−1)=−8

Final Answer

(lim_x→4)((x2−16)/(4−x))=−8

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