Find the limits as x → ∞ and as x → −∞.

Question

y = f(x) = (2 − x)(1 + x)^2(1 − x)^4

∞ = ?

−∞ = ?

Answer 1

The first step would be to expand f(x) and combine like terms,

f(x) = (2 − x)(1 + x)^2(1 − x)^4

f(x) = -x⁷+4x⁶-3x⁵-6x⁴+9x³-5x+2

From here, we can conclude that the term with the highest degree, in this case -x⁷ , will "win" when we allow x to approach ∞ or −∞.

Therefore,

The limit of f(x) approaches −∞ as x approaches ∞

and

The limit of f(x) approaches ∞ as x approaches -∞