Evaluate the Limit

Problem

(lim_x→4)((x−4)/(x2−16))

Solution

1. Identify the form of the limit by substituting x=4 into the expression.

(4−4)/(4−16)=0/0

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

x2−16=(x−4)*(x+4)

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

(x−4)/((x−4)*(x+4))=1/(x+4)

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

(lim_x→4)(1/(x+4))=1/(4+4)

5. Calculate the final numerical value.

1/(4+4)=1/8

Final Answer

(lim_x→4)((x−4)/(x2−16))=1/8

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