Evaluate the Limit

Problem

(lim_x→16)((x2+3*x−304)/(x2−256))

Solution

1. Substitute the value x=16 into the expression to check for an indeterminate form.

(16+3*(16)−304)/(16−256)=(256+48−304)/(256−256)=0/0

2. Factor the numerator x2+3*x−304 by finding two numbers that multiply to −304 and add to 3

x2+3*x−304=(x−16)*(x+19)

3. Factor the denominator x2−256 using the difference of squares formula a2−b2=(a−b)*(a+b)

x2−256=(x−16)*(x+16)

4. Simplify the expression by canceling the common factor (x−16) from the numerator and the denominator.

((x−16)*(x+19))/((x−16)*(x+16))=(x+19)/(x+16)

5. Evaluate the limit by substituting x=16 into the simplified expression.

(lim_x→16)((x+19)/(x+16))=(16+19)/(16+16)=35/32

Final Answer

(lim_x→16)((x2+3*x−304)/(x2−256))=35/32

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