Evaluate the Limit

Problem

(lim_x→−24)((√(,x^2+49)−25)/(x+24))

Solution

1. Identify the indeterminate form by substituting x=−24 into the expression.

(√(,(−24)^2+49)−25)/(−24+24)=(√(,576+49)−25)/0

(√(,625)−25)/0=(25−25)/0=0/0

2. Rationalize the numerator by multiplying the numerator and the denominator by the conjugate √(,x^2+49)+25

(√(,x^2+49)−25)/(x+24)⋅(√(,x^2+49)+25)/(√(,x^2+49)+25)

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a^2−b^2

((√(,x^2+49))^2−25^2)/((x+24)*(√(,x^2+49)+25))

(x^2+49−625)/((x+24)*(√(,x^2+49)+25))

4. Simplify the numerator by combining like terms.

(x^2−576)/((x+24)*(√(,x^2+49)+25))

5. Factor the numerator using the difference of squares x^2−576=(x−24)*(x+24)

((x−24)*(x+24))/((x+24)*(√(,x^2+49)+25))

6. Cancel the common factor (x+24) from the numerator and the denominator.

(x−24)/(√(,x^2+49)+25)

7. Evaluate the limit by substituting x=−24 into the simplified expression.

(−24−24)/(√(,(−24)^2+49)+25)

(−48)/(√(,625)+25)

(−48)/(25+25)=(−48)/50

8. Reduce the fraction to its simplest form.

−24/25

Final Answer

(lim_x→−24)((√(,x^2+49)−25)/(x+24))=−24/25

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