The end behavior of the function f(x) = (x - 5)/(5x^3 - 2x^2 + 3) is f(x) ⇒ 0 as x ⇒ ∝ and f(x) ⇒ 0 as x ⇒ -∝
<h3>How to determine the end behaviors of the function?</h3>
The function is given as:
f(x) = (x - 5)/(5x^3 - 2x^2 + 3)
Next, we determine the limits of the function.
This is represented as:
lim x ⇒ ∝ and -∝ of f(x) = (x - 5)/(5x^3 - 2x^2 + 3)
For lim x ⇒ ∝, we have:
f(∝) = (∝ - 5)/(5(∝)^3 - 2(∝)^2 + 3)
Evaluate the exponents
f(∝) = (∝ - 5)/(5(∝) - 2(∝) + 3)
Evaluate the products
f(∝) = (∝ - 5)/(∝ - ∝ + 3)
Evaluate the sum and the difference
f(∝) = ∝/∝
Evaluate the quotient
f(∝) = 0
For lim x ⇒ -∝, we have:
f(-∝) = (-∝ - 5)/(5(-∝)^3 - 2(-∝)^2 + 3)
Evaluate the exponents
f(-∝) = (-∝ - 5)/(5(-∝) - 2(∝) + 3)
Evaluate the products
f(-∝) = (-∝ - 5)/(-∝ - ∝ + 3)
Evaluate the sum and the difference
f(-∝) = -∝/-∝
Evaluate the quotient
f(-∝) = 0
So, we have:
f(x) ⇒ 0 as x ⇒ ∝ and f(x) ⇒ 0 as x ⇒ -∝
Hence, the end behavior of the function f(x) = (x - 5)/(5x^3 - 2x^2 + 3) is f(x) ⇒ 0 as x ⇒ ∝ and f(x) ⇒ 0 as x ⇒ -∝
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