A)) Choose the correct statement. \( \lim _{x \rightarrow 6} \frac{1}{(x-6)^{7}}=-\infty \) \( \lim _{x \rightarrow 6} \frac{1}{(x-6)^{7}}=\infty \) \( \lim _{x \rightarrow 6} \frac{1}{(x-6)^{7}} \) is neither \( -\infty \) nor \( \infty \)

When we evaluate \( \lim_{x \rightarrow 6} \frac{1}{(x-6)^{7}} \), we notice that as \( x \) approaches \( 6 \), \( (x-6)^{7} \) approaches \( 0 \). Since the exponent is odd, when \( x \) approaches \( 6 \) from the left, \( (x-6) \) is negative, resulting in a negative denominator—leading the whole fraction to approach \( -\infty \). Conversely, approaching from the right, \( (x-6) \) is positive, making the fraction approach \( \infty \). So the correct statement here is that the limit diverges based on the direction of approach. You can see how limits can really capture the essence of a function's behavior, and it's fascinating how a single point can lead to such different outcomes!