Evaluate the Limit

Problem

(lim_x→5)((x−5)/(x2−25))

Solution

1. Identify the indeterminate form by substituting x=5 into the expression, which results in 0/0

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

x2−25=(x−5)*(x+5)

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

(x−5)/((x−5)*(x+5))=1/(x+5)

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

1/(5+5)=1/10

Final Answer

(lim_x→5)((x−5)/(x2−25))=1/10

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