QuestionYou 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, $c$. Remember to type "infinity" for $\infty$, "-infinity" for $-\infty$, and "NA" if the limit does not exist. Enter the results and I will apply. the passcode." $$\lim _{x \rightarrow c^{+}} \frac{18}{(x-c)^{3}}=$$ $\square$ $$\lim _{x \rightarrow c^{-}} \frac{18}{(x-c)^{3}}=$$ $\square$
Explain, in your own words and with your own work, how you arrived at this result. Be sure to explain using calculus concepts to best support the work of the game design team.

Studdy Solution
Consider the limit as $x$ approaches $c^-$: $$\lim_{x \to c^-} \frac{18}{(x-c)^3}$$
As $x$ approaches $c$ from the left, $(x-c)$ is a small negative number. Therefore, $(x-c)^3$ is a small negative number raised to the third power, which remains negative. Since the numerator is positive (18), the entire fraction becomes very large and negative as $x$ approaches $c^-$.
Thus, the limit is: $$\lim_{x \to c^-} \frac{18}{(x-c)^3} = -\infty$$
The results for the limits are: - For $\lim_{x \to c^+} \frac{18}{(x-c)^3}$, the result is "infinity". - For $\lim_{x \to c^-} \frac{18}{(x-c)^3}$, the result is "-infinity".