Evaluate the Limit

Problem

(lim_x→169)((√(,x)−13)/(x−169))

Solution

1. Identify the indeterminate form by substituting x=169 into the expression, which results in (13−13)/(169−169)=0/0

2. Factor the denominator using the difference of squares formula, a2−b2=(a−b)*(a+b) where x=(√(,x))2 and 169=13

3. Rewrite the expression with the factored denominator.

(√(,x)−13)/((√(,x)−13)*(√(,x)+13))

4. Simplify the fraction by canceling the common factor (√(,x)−13) from the numerator and the denominator.

1/(√(,x)+13)

5. Evaluate the limit by substituting x=169 into the simplified expression.

1/(√(,169)+13)

6. Calculate the final numerical value.

1/(13+13)=1/26

Final Answer

(lim_x→169)((√(,x)−13)/(x−169))=1/26

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