Questions: Evaluate the limit. Answer exactly. lim x→-3 (x^2-9)/(x+3)=

Solution Steps

To evaluate the limit \(\lim _{x \rightarrow-3} \frac{x^{2}-9}{x+3}\), we can simplify the expression by factoring the numerator. The expression \(x^2 - 9\) is a difference of squares and can be factored as \((x - 3)(x + 3)\). This allows us to cancel the common factor of \(x + 3\) in the numerator and the denominator, simplifying the expression. After simplification, we can directly substitute \(x = -3\) to find the limit.

We start with the limit expression: \[ \lim _{x \rightarrow -3} \frac{x^{2}-9}{x+3} \] The numerator \(x^2 - 9\) can be factored as a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3) \] Thus, we can rewrite the limit as: \[ \lim _{x \rightarrow -3} \frac{(x - 3)(x + 3)}{x + 3} \]

Next, we can cancel the common factor \(x + 3\) in the numerator and the denominator, provided \(x \neq -3\): \[ \lim _{x \rightarrow -3} (x - 3) \]

Now, we can directly substitute \(x = -3\) into the simplified expression: \[ -3 - 3 = -6 \]

Final Answer