use the properties of limits to help decide whether the limit exists. if the limit exists, find its value…

use the properties of limits to help decide whether the limit exists. if the limit exists, find its value. lim x→ - 3 x² - 9 / x + 3. simplify the rational expression. x² - 9 / x + 3 =

use the properties of limits to help decide whether the limit exists. if the limit exists, find its value. lim x→ - 3 x² - 9 / x + 3. simplify the rational expression. x² - 9 / x + 3 =

Answer

Explanation:

Step1: Factor the numerator

We know that $x^{2}-9=(x + 3)(x - 3)$ by the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ where $a=x$ and $b = 3$. So, $\frac{x^{2}-9}{x + 3}=\frac{(x + 3)(x - 3)}{x+3}$.

Step2: Simplify the rational expression

Cancel out the common factor $(x + 3)$ in the numerator and the denominator. We get $\frac{(x + 3)(x - 3)}{x+3}=x - 3$ for $x\neq - 3$.

Step3: Find the limit

Now, we find $\lim_{x\rightarrow - 3}\frac{x^{2}-9}{x + 3}=\lim_{x\rightarrow - 3}(x - 3)$. Substitute $x=-3$ into $x - 3$. We have $-3-3=-6$.

Answer:

-6