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 will apply the
passcode”

Ask by Boyd Alexander. 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 will apply the
passcode”

Ask by Boyd Alexander. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

We start by letting $$ u = x-c $$. Then the expression becomes:

$$
\frac{27}{(x-c)^5} = \frac{27}{u^5}.
$$

1. When $$ x \rightarrow c^- $$ (approaching from the left), we have $$ u < 0 $$ with $$ u \rightarrow 0 $$. Since the exponent is 5 (an odd number), $$ u^5 $$ remains negative. Therefore, the denominator becomes a very small negative number, meaning

$$
\frac{27}{u^5} \rightarrow -\infty.
$$

2. When $$ x \rightarrow c^+ $$ (approaching from the right), we have $$ u > 0 $$ with $$ u \rightarrow 0 $$. Here, $$ u^5 $$ is a very small positive number. Thus,

$$
\frac{27}{u^5} \rightarrow \infty.
$$

The final answers are:

$$
\lim _{x \rightarrow c^{-}} \frac{27}{(x-c)^{5}} = -\infty
$$

$$
\lim _{x \rightarrow c^{+}} \frac{27}{(x-c)^{5}} = \infty.
$$
$$ \lim _{x \rightarrow c^{-}} \frac{27}{(x-c)^{5}} = -\infty $$ $$ \lim _{x \rightarrow c^{+}} \frac{27}{(x-c)^{5}} = \infty $$

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