Evaluate the Limit

Problem

(lim_t→−3)((t2−9)/(2*t2+7*t+3))

Solution

1. Substitute the value t=−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)

t2−9=(t−3)*(t+3)

3. Factor the quadratic expression in the denominator by finding two numbers that multiply to 6 (from 2×3 and add to 7

2*t2+7*t+3=(2*t+1)*(t+3)

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

(lim_t→−3)(((t−3)*(t+3))/((2*t+1)*(t+3)))=(lim_t→−3)((t−3)/(2*t+1))

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

(−3−3)/(2*(−3)+1)=(−6)/(−6+1)=(−6)/(−5)

Final Answer

(lim_t→−3)((t2−9)/(2*t2+7*t+3))=6/5

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