The First Gate.
Upon arriving at the first gate, you ask your computer to translate the riddle. “These appear to be limits”, your
computer replies. “If you provide me with the answer to each of below questions, I will compile, translate, and
communicate the passcode to the gate. There is note here that the parameter represents a real number such
that .” Your computer also reminds you to type “infinity” for , “-infinity” for , and “NA” if the
limit does not exist. You grab your notebook and pencil and begin.

Question

The First Gate.
Upon arriving at the first gate, you ask your computer to translate the riddle. “These appear to be limits”, your
computer replies. “If you provide me with the answer to each of below questions, I will compile, translate, and
communicate the passcode to the gate. There is note here that the parameter represents a real number such
that .” Your computer also reminds you to type “infinity” for , “-infinity” for , and “NA” if the
limit does not exist. You grab your notebook and pencil and begin.

UpStudy AI · Accepted Answer

1. For $$ x ightarrow a^{-} $$:
   - As $$ x $$ approaches $$ a $$ from the left, we have $$ x < a $$. Thus, the denominator $$ x-a $$ is a small negative number.
   - The numerator $$ x+6 $$ evaluated at $$ x $$ near $$ a $$ is approximately $$ a+6 $$, which is positive since $$ 0 < a < 6 $$.
   - Therefore, the result is a positive number divided by a small negative number, which gives $$-\infty$$.

2. For $$ x ightarrow a^{+} $$:
   - As $$ x $$ approaches $$ a $$ from the right, we have $$ x > a $$. Thus, the denominator $$ x-a $$ is a small positive number.
   - The numerator $$ x+6 $$ remains approximately $$ a+6 $$, which is positive.
   - Hence, the result is a positive number divided by a small positive number, which gives $$+\infty$$.

3. For $$ x ightarrow a $$:
   - The left-hand limit is $$-\infty$$ and the right-hand limit is $$+\infty$$.
   - Since the two one-sided limits are not equal, the overall limit does not exist. We denote this by "NA".

Thus, we have:
- $$ \lim_{x ightarrow a^{-}} \frac{x+6}{x-a} = -\infty $$
- $$ \lim_{x ightarrow a^{+}} \frac{x+6}{x-a} = \infty $$
- $$ \lim_{x ightarrow a} \frac{x+6}{x-a} = 	ext{NA} $$
- As $$ x $$ approaches $$ a $$ from the left, the limit is $$-\infty$$. - As $$ x $$ approaches $$ a $$ from the right, the limit is $$+\infty$$. - The overall limit as $$ x $$ approaches $$ a $$ does not exist (NA).

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