Example 4

If ε ≠ 0, the quotient

is infinite.

We first note that the denominator 5ε⁴ + ε³ is not zero because it can be written as a product of nonzero factors,

5ε⁴ + ε³ = ε · ε · ε · (5ε + 1).

When we cancel ε² from the numerator and denominator we get

We see in turn that:

ε² - ε + 2 is finite but not infinitesimal,

5ε² + ε is infinitesimal,

is infinite.