Find \(\lim\limits_{x \to 3} (\frac{x^{3} + x^{2} - 12x}{x^{2} - 9})\)
Question 1 Report
Find \(\lim\limits_{x \to 3} (\frac{x^{3} + x^{2} - 12x}{x^{2} - 9})\)
Answer Details
To evaluate the limit, we can substitute the value of x into the expression and see what we get. However, in this case, x = 3 would make the denominator 0, and division by 0 is undefined. This suggests that we need to simplify the expression first.
We can factor the numerator and denominator, then simplify the expression:
\[\lim\limits_{x \to 3} \frac{x^{3} + x^{2} - 12x}{x^{2} - 9} = \lim\limits_{x \to 3} \frac{x(x-3)(x+4)}{(x+3)(x-3)} = \lim\limits_{x \to 3} \frac{x(x+4)}{x+3}\]
Now, we can substitute x = 3 into the simplified expression:
\[\frac{3(3+4)}{3+3} = \frac{21}{6} = \frac{7}{2}\]
Therefore, the answer is \(\frac{7}{2}\).
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