The end behavior of the graph of f(x) = x3(x + 3)(–5x + 1) using limits is as x ⇒ ∝, f(x) ⇒ -∝ and x ⇒ -∝, f(x) ⇒ ∝
How to determine the end behavior of the graph of f(x) = x3(x + 3)(–5x + 1) using limits?
The equation of the function is given as:
f(x) = x^3(x + 3)(–5x + 1)
Using limits, we have:
As x approaches positive infinity, the function becomes
f(∝) = (∝)^3(∝ + 3)(–5(∝) + 1)
Evaluate the products and exponents
f(∝) = (∝)(∝)(–∝ + 1)
Evaluate the difference
f(∝) = (∝)(∝)(–∝)
Evaluate the product
f(∝) = -∝
This means that as x ⇒ ∝, f(x) ⇒ -∝
Using limits, we have:
As x approaches negative infinity, the function becomes
f(-∝) = (-∝)^3(-∝ + 3)(–5(-∝) + 1)
Evaluate the products and exponents
f(-∝) = (-∝)(-∝)(∝ + 1)
Evaluate the sum
f(-∝) = (-∝)(-∝)(∝)
Evaluate the product
f(-∝) = ∝
This means that as x ⇒ -∝, f(x) ⇒ ∝
Hence, the end behavior of the graph of f(x) = x3(x + 3)(–5x + 1) using limits is as x ⇒ ∝, f(x) ⇒ -∝ and x ⇒ -∝, f(x) ⇒ ∝
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