Questions: Evaluate the following limit. Enter an exact answer. lim (x -> -8) (x^2 - 64) / (x + 8) = Number T

Solution Steps

To evaluate the limit, we can simplify the expression by factoring the numerator. The numerator \(x^2 - 64\) is a difference of squares, which can be factored into \((x - 8)(x + 8)\). This allows us to cancel the \((x + 8)\) term in the numerator and denominator, simplifying the expression. After simplification, we can directly substitute \(x = -8\) to find the limit.

We start with the limit expression: \[ \lim_{x \rightarrow -8} \frac{x^2 - 64}{x + 8} \] The numerator \(x^2 - 64\) can be factored as a difference of squares: \[ x^2 - 64 = (x - 8)(x + 8) \] Thus, we can rewrite the limit as: \[ \lim_{x \rightarrow -8} \frac{(x - 8)(x + 8)}{x + 8} \]

We can cancel the \((x + 8)\) term in the numerator and denominator, provided \(x \neq -8\): \[ \lim_{x \rightarrow -8} (x - 8) \]

Now, we substitute \(x = -8\) into the simplified expression: \[ -8 - 8 = -16 \]

Final Answer