Evaluate the Limit

Problem

(lim_x→−12)((√(,x2+25)−13)/(x+12))

Solution

1. Identify the indeterminate form by substituting x=−12 into the expression, which results in (√(,(−12)2+25)−13)/(−12+12)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate expression √(,x2+25)+13

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

((√(,x2+25)−13)*(√(,x2+25)+13))/((x+12)*(√(,x2+25)+13))=((x2+25)−169)/((x+12)*(√(,x2+25)+13))

4. Simplify the numerator by combining the constants.

(x2−144)/((x+12)*(√(,x2+25)+13))

5. Factor the numerator using the difference of squares x2−144=(x−12)*(x+12)

((x−12)*(x+12))/((x+12)*(√(,x2+25)+13))

6. Cancel the common factor (x+12) from the numerator and the denominator.

(x−12)/(√(,x2+25)+13)

7. Evaluate the limit by substituting x=−12 into the simplified expression.

(−12−12)/(√(,(−12)2+25)+13)=(−24)/(√(,169)+13)

(−24)/(13+13)=(−24)/26

8. Reduce the fraction to its simplest form.

−12/13

Final Answer

(lim_x→−12)((√(,x2+25)−13)/(x+12))=−12/13

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