Evaluate lim x → ∞ e − x 2 ∫ x x + ln ⁡ ( x ) / x e − t 2 d t

Question

Evaluate       lim          x      →      ∞            e          −              x        2                  ∫    x          x      +      ln      ⁡      (      x      )              /            x            e          −              t        2              d  t

soosenhc · Accepted Answer

First, we&#039;ll use a limit law, saying the limit of a product is a product of the limits.      lim          x      →      ∞            e          −              x        2              ⋅      lim          x      →      ∞            ∫    x          x      +                        ln          ⁡          (          x          )                x                  e          −              t        2              d  tLet&#039;s evaluate the first limit.      lim          x      →      ∞            e          −              x        2              =      lim          x      →      ∞            1          e                        x          2                    As x approaches   ∞,       e                  x        2             also approaches   ∞, therefore       1          e                        x          2                     approaches 0, therefore one of the limits calculated is 0.However, we&#039;re not done. If the limit of the integral is   ∞, we would get an indeterminate form. By monotonicity,      |              ∫      x              x        +                              ln            ⁡            (            x            )                    x                            e              −                  t          2                      d    t        |    ≤  (  x  +            ln      ⁡      (      x      )        x    −  x  )      e          −              x        2              =            ln      ⁡      (      x      )        x        e          −              x        2            , which if we evaluate as x approaches   ∞, we get   0  ⋅  0  =  0, so we didn&#039;t have to use L&#039;Hôpital&#039;s rule.Hope this helps!

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