Math  /  Calculus

QuestionIs there a number a such that the following limit exists? (If an answer does not exist, enter DNE.) limx23x2+ax+a+9x2+x2\lim _{x \rightarrow-2} \frac{3 x^{2}+a x+a+9}{x^{2}+x-2}
Find the value aa. a=7a=-7
Evaluate the limit. 55

Studdy Solution
Simplify the expression by canceling the common factor (x+2) (x + 2) :
3(x+2)(x+5)(x+2)(x1)=3(x+5)x1 \frac{3(x + 2)(x + 5)}{(x + 2)(x - 1)} = \frac{3(x + 5)}{x - 1}
Evaluate the limit as x2 x \rightarrow -2 :
limx23(x+5)x1=3(2+5)21 \lim_{x \rightarrow -2} \frac{3(x + 5)}{x - 1} = \frac{3(-2 + 5)}{-2 - 1} =3×33=3 = \frac{3 \times 3}{-3} = -3
The limit does not exist for a=7 a = -7 . The correct value of a a should be 21, and the evaluated limit is 3 \boxed{-3} .

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