Indeterminate form 0 x infinity

We will suppose that limx→+∞f(x)=0 and limx→+∞g(x)=±∞, then we will have that limx→+∞f(x)⋅g(x)=0⋅±(∞).

In other words, we are wondering what function goes more rapidly to its limit, f(x) to zero or g(x) to infinity.

To solve this type of indeterminate form we will do a simple step: limx→+∞f(x)⋅g(x)=limx→+∞11f(x)⋅g(x)=limx→+∞g(x)1f(x)=±∞±∞ and we will solve the limit.

Let's see some examples:

Example

  1. limx→+∞2xx3−1⋅ln⁡x=0⋅(+∞)⇒limx→+∞2xx3−1⋅ln⁡x=limx→+∞2xln⁡xx3−1=

=limx→+∞2x⋅ln⁡xx3=limx→+∞2⋅ln⁡xx2=0

  1. limx→+∞x−x⋅2x=limx→+∞2xxx=0

  2. limx→+∞ln⁡xx+1⋅−x2−1ln⁡x−1=limx→+∞ln⁡x⋅(−x2−1)(x+1)⋅(ln⁡x−1)=limx→+∞−x=−∞