Evaluate the Limit

Problem

(lim_x→−3)((x2−9)/(2*x2+7*x+3))

Solution

1. Substitute the value x=−3 into the expression to check for an indeterminate form.

((−3)2−9)/(2*(−3)2+7*(−3)+3)=(9−9)/(18−21+3)=0/0

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

x2−9=(x−3)*(x+3)

3. Factor the denominator by finding two numbers that multiply to 2⋅3=6 and add to 7

2*x2+7*x+3=(2*x+1)*(x+3)

4. Simplify the expression by canceling the common factor (x+3) from the numerator and the denominator.

((x−3)*(x+3))/((2*x+1)*(x+3))=(x−3)/(2*x+1)

5. Evaluate the limit by substituting x=−3 into the simplified expression.

(lim_x→−3)((x−3)/(2*x+1))=(−3−3)/(2*(−3)+1)

6. Calculate the final numerical value.

(−6)/(−6+1)=(−6)/(−5)=6/5

Final Answer

(lim_x→−3)((x2−9)/(2*x2+7*x+3))=6/5

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