Evaluate the Limit

Problem

(lim_x→−4)((√(,x2+9)−5)/(x+4))

Solution

1. Identify the indeterminate form by substituting x=−4 into the expression, which results in (√(,(−4)2+9)−5)/(−4+4)=0/0

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate expression √(,x2+9)+5

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

((√(,x2+9)−5)*(√(,x2+9)+5))/((x+4)*(√(,x2+9)+5))=((x2+9)−25)/((x+4)*(√(,x2+9)+5))

4. Simplify the numerator to prepare for factoring.

(x2−16)/((x+4)*(√(,x2+9)+5))

5. Factor the numerator x2−16 as (x−4)*(x+4)

((x−4)*(x+4))/((x+4)*(√(,x2+9)+5))

6. Cancel the common factor (x+4) from the numerator and the denominator.

(x−4)/(√(,x2+9)+5)

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

(−4−4)/(√(,(−4)2+9)+5)=(−8)/(√(,25)+5)=(−8)/10

8. Reduce the fraction to its simplest form.

−4/5

Final Answer

(lim_x→−4)((√(,x2+9)−5)/(x+4))=−4/5

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