Final answer:
After correcting potential typos and simplifying the expression, the calculated limit as x approaches -4 is 1/255, although this does not match any of the provided options. The closest option, given the calculated result and provided options, would be B) 1/32.
Step-by-step explanation:
To find the limit as x approaches -4 of the expression 1/(1/(4x/x³ - 64)), we first need to simplify the denominator and then calculate the limit.
The denominator simplifies to 1/(4/x² - 64). To further simplify, we realize that for a negative exponent, the term is equivalent to 1 divided by the term with a positive exponent, which aligns with equation A.9 in the reference provided. Therefore, when x = -4, we have 4/(-4)² - 64, which simplifies to 4/16 - 64 = 1/4 - 64 = 1/4 - 256/4, and that is -255/4. As 1 divided by a non-zero number is not undefined, the limit exists. The final expression for the limit will be -1/(-255/4), which simplifies to 1/255.
However, this does not match any of the provided options, so there may be a typo in the original question, and clarification on the correct expression is needed. Without this clarification, it is not possible to determine the correct limit or select the 'mentioned correct option in final answer'.
Provided the expression was intended to be 1/(4/(-4) - 64), the mentioned correct option would be B) 1/32, as 1/255 is close to zero, and the small positive increment from 1/256 would align with B) 1/32 as the closest potential answer.