Suppose the total cost, C(q), of producing a quantity q of a product is
given by the equation
C(q) = 5000 + 5q
The average cost per unit quantity, A(q), equals the total cost, C(q),
divided by the quantity produced, q. Find the limiting value of the average
cost per unit as q tends to infinity. In other words find
lim--> infinite A(q)
Well, as q tends to infinity, the 5q term becomes the dominant factor in the equation. So, we can simplify it as:
C(q) ≈ 5q
Now, to find the limiting value of the average cost per unit as q tends to infinity, we divide both sides by q:
A(q) ≈ (5q) / q
A(q) ≈ 5
Voila! The limiting value of the average cost per unit as q tends to infinity is 5. But don't worry, you won't actually have to pay millions for one unit of that product! It's just a mathematical approximation.
To find the limiting value of the average cost per unit as q tends to infinity, we need to evaluate the limit of A(q) as q approaches infinity.
The average cost per unit, A(q), is defined as the total cost, C(q), divided by the quantity produced, q. So we have:
A(q) = C(q) / q
Given that C(q) = 5000 + 5q, we can substitute this into the equation for A(q):
A(q) = (5000 + 5q) / q
To evaluate the limit of A(q) as q approaches infinity, we can divide both the numerator and the denominator by q:
A(q) = (5000/q + 5)
As q approaches infinity, the term 5000/q becomes very small compared to 5. Thus, the limiting value of A(q) is simply 5:
lim(q->∞)A(q) = 5
Therefore, the average cost per unit approaches 5 as the quantity produced tends to infinity.
To find the limiting value of average cost per unit as q tends to infinity, we need to determine the behavior of A(q) as q becomes larger and larger.
The average cost per unit, A(q), is calculated by dividing the total cost, C(q), by the quantity produced, q. So, we can express A(q) as:
A(q) = C(q) / q
Given that C(q) = 5000 + 5q, we can substitute this into the equation for A(q):
A(q) = (5000 + 5q) / q
To find the limiting value of A(q) as q tends to infinity, we can take the limit of A(q) as q approaches infinity.
lim(q -> infinity) A(q) = lim(q -> infinity) [(5000 + 5q) / q]
To simplify the expression, we can divide the numerator and the denominator by q:
lim(q -> infinity) [(5000/q + 5)]
As q approaches infinity, the term 5000/q tends to 0 because q is becoming infinitely large. Therefore, we are left with:
lim(q -> infinity) 5
The limiting value of A(q) as q tends to infinity is 5.
So, the average cost per unit when q tends to infinity is 5.