Evaluate the Limit

Problem

(lim_x→4)((x2+3*x−28)/(x2−16))

Solution

1. Substitute the value x=4 into the expression to check for an indeterminate form.

(4+3*(4)−28)/(4−16)=(16+12−28)/(16−16)=0/0

2. Factor the numerator by finding two numbers that multiply to −28 and add to 3

x2+3*x−28=(x+7)*(x−4)

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

x2−16=(x−4)*(x+4)

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

((x+7)*(x−4))/((x−4)*(x+4))=(x+7)/(x+4)

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

(4+7)/(4+4)=11/8

Final Answer

(lim_x→4)((x2+3*x−28)/(x2−16))=11/8

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