The improper integral is defined for positive ε; the intended limit is ε↓0. Put x=εt to get $\int_0^\infty e^{-t}[\cos(3\varepsilon t)+\varepsilon^2t^2+\sqrt{4+\varepsilon t}]dt$. Dominated convergence for 0<ε≤1 gives (1+0+2)∫e⁻ᵗdt=**3, C**. The source omits the positive-side qualifier; for negative ε the improper integral diverges.
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