The Third Gate
You arrive at the third gate and look for the inscription. Your computer translates the following:
“This passcode consists of the two below limits which include an arbitary real number, . Remember to type
“infinity” for , “-infinity” for , and “NA” if the limit does not exist Enter the results and I will apply the
passcode.”

Ask by Clark Byrd. in the United States

Mar 28,2025

Question

The Third Gate
You arrive at the third gate and look for the inscription. Your computer translates the following:
“This passcode consists of the two below limits which include an arbitary real number, . Remember to type
“infinity” for , “-infinity” for , and “NA” if the limit does not exist Enter the results and I will apply the
passcode.”

Ask by Clark Byrd. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

The first limit is:

$$
\lim_{x \rightarrow c} \frac{27}{(x-c)^5}
$$

As $$ x $$ approaches $$ c $$, the denominator $$ (x-c)^5 $$ tends to $$ 0 $$. However, the behavior is different when approaching from the left ($$ x < c $$) versus the right ($$ x > c $$) because the power $$ 5 $$ is odd:

- For $$ x \rightarrow c^{-} $$: $$ (x-c) $$ is negative, thus $$ (x-c)^5 $$ is negative, so the fraction tends to $$-\infty$$.
- For $$ x \rightarrow c^{+} $$: $$ (x-c) $$ is positive, hence $$ (x-c)^5 $$ is positive, so the fraction tends to $$ \infty $$.

Since the left-hand and right-hand limits differ, the overall limit does not exist. Therefore, the answer is: NA

The second limit is:

$$
\lim_{x \rightarrow c^{+}} \frac{27}{(x-c)^5}
$$

Here, as $$ x $$ approaches $$ c $$ from the right, $$ (x-c) $$ is positive, making $$ (x-c)^5 $$ a small positive number. Dividing $$ 27 $$ by a very small positive number results in a value that grows without bound, namely $$ \infty $$. Thus, the answer is: infinity
The first limit does not exist (NA), and the second limit is infinity.

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