To determine the horizontal asymptote of the average cost function [tex]\(\bar{c}(x) = \frac{200,000 + 450x}{x}\)[/tex], let's analyze the behavior of the function as [tex]\(x\)[/tex] approaches infinity.
1. Rewrite the function:
[tex]\[
\bar{c}(x) = \frac{200,000 + 450x}{x}
\][/tex]
2. Separate the terms in the numerator:
[tex]\[
\bar{c}(x) = \frac{200,000}{x} + \frac{450x}{x}
\][/tex]
3. Simplify the expression:
[tex]\[
\bar{c}(x) = \frac{200,000}{x} + 450
\][/tex]
4. Analyze the terms as [tex]\(x \to \infty\)[/tex]:
- The term [tex]\(\frac{200,000}{x}\)[/tex] approaches 0 because you are dividing a constant by a larger and larger value.
- The term [tex]\(450\)[/tex] remains constant, as it is independent of [tex]\(x\)[/tex].
5. Sum the limits:
[tex]\[
\lim_{x \to \infty} \left( \frac{200,000}{x} + 450 \right) = 0 + 450
\][/tex]
Therefore, the limit of [tex]\(\bar{c}(x)\)[/tex] as [tex]\(x\)[/tex] approaches infinity is [tex]\(450\)[/tex].
Putting it all together, the horizontal asymptote of [tex]\(\bar{c}(x)\)[/tex] is:
[tex]\[
y = 450
\][/tex]
So, the horizontal asymptote of the average cost function [tex]\(\bar{c}(x)\)[/tex] is [tex]\(y = 450\)[/tex].
