1. A company has to spend $$ \$ 20,000 $$ in order to properly run. Additionally, this company has costs of $$ \$ 6 $$ per product for the first 5,000 produced units. If the company produces more than 5,000 units, the additional production cost for a unit is $$ \$ 4 $$. Find the production cost function $$ p(x) $$ for $$ x $$ units, and find the following values if they exist: $$ \lim _{x \rightarrow 5,000^{-}} p(x), \lim _{x \rightarrow 5,000^{+}} p(x), \lim _{x \rightarrow 5,000} p(x) $$

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The production cost function $$ p(x) $$ is: $$ p(x) = \begin{cases} 20000 + 6x & \text{if } x \leq 5000 \ 4x + 30000 & \text{if } x > 5000 \end{cases} $$ The limits are: $$ \lim_{x \to 5000^{-}} p(x) = 50000, \quad \lim_{x \to 5000^{+}} p(x) = 50000, \quad \lim_{x \to 5000} p(x) = 50000 $$

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