Final answer:
The improper integral ∫[infinity]₀ (e⁻²ˣ/⁴-e⁻ˣ²)/(x) dx is divergent.
Step-by-step explanation:
In order to evaluate the improper integral ∫[infinity]₀ (e⁻²ˣ/⁴-e⁻ˣ²)/(x) dx, we need to use the limit comparison test. This test states that if two integrals have the same upper and lower limits, and the limit of the ratio of their integrands as x approaches infinity is finite, then they either both converge or both diverge.
We will compare the integrand (e⁻²ˣ/⁴-e⁻ˣ²)/(x) with a known function, the harmonic series, which is given by 1/x. The ratio of the two functions is (e⁻²ˣ/⁴-e⁻ˣ²)/(1). As x approaches infinity, this ratio approaches zero, so the two integrals must have the same behavior. The harmonic series is known to diverge, so the integral ∫[infinity]₀ (e⁻²ˣ/⁴-e⁻ˣ²)/(x) dx must also diverge.