Evaluate the Limit

Problem

(lim_x→4)((x2−16)/(x2−5*x+4))

Solution

1. Identify the indeterminate form by substituting x=4 into the expression, which results in (4−16)/(4−5*(4)+4)=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. Factor the denominator by finding two numbers that multiply to 4 and add to −5

x2−5*x+4=(x−4)*(x−1)

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

((x−4)*(x+4))/((x−4)*(x−1))=(x+4)/(x−1)

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

(4+4)/(4−1)=8/3

Final Answer

(lim_x→4)((x2−16)/(x2−5*x+4))=8/3

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