For any α ∈ R determine the limit lim &gt; δ → 0 ∫ 0 2 1 − x δ + x α d x in [ − ∞ , + ∞ ] .

Question

For any   α  ∈      R   determine the limit      lim                                                      &amp;gt;                                                          δ              →              0                                                ∫    0    2              1      −      x              δ      +              x                  α                      d  x    in   [  −  ∞  ,  +  ∞  ]  .

partyjnopp9wa · Accepted Answer

Denote for   α  ∈      R   and   δ  &amp;gt;  0      f          α      ,      δ        (  x  )  =            1      −      x              δ      +              x                  α                       and       F          α        (  x  )  =            1      −      x              x              α              .For   x  &amp;gt;  0, you have  |      f          α      ,      δ        (  x  )  |  ≤  |      F          α        (  x  )  |and      lim                                                      &amp;gt;                                                          δ              →              0                                                f          α      ,      δ        (  x  )  =      F          α        (  x  )  .As       ∫    0    2    |      F          α        (  x  )  |     d  x is convergent for   α  &amp;lt;  1, you can apply Dominated Convergent Theorem - DCT in that case and get      lim                                                      &amp;gt;                                                          δ              →              0                                                ∫    0    2              1      −      x              δ      +              x                  α                      d  x  =      ∫    0    2        F          α        (  x  )     d  x  .So let&#039;s now suppose that   α  ≥  1You&#039;ll easily prove that      ∫    1    2        f          α      ,      δ        (  x  )     d  xis bounded by 3 for   (  α  ,  δ  )  ∈  [  1  ,  ∞  )  ×  (  0  ,  ∞  ). This implies the implication      lim                                                      &amp;gt;                                                          δ              →              0                                                ∫    0    1        f          α      ,      δ        (  x  )     d  x  =  ∞    ⟹        lim                                                      &amp;gt;                                                          δ              →              0                                                ∫    0    2        f          α      ,      δ        (  x  )     d  x  =  ∞  .Now for   0  &amp;lt;  x  ≤  1 (and   α  ≥  1 as assumed), we have  0  ≤            1      −      x              δ      +      x        ≤      f          α      ,      δ        (  x  )  .We get the desired result as      ∫    0    1              1      −      x              δ      +      x           d  x  =  −  1  +  (  1  +  δ  )  ln  ⁡      (    1    +          1      δ        )  and      lim                                                      &amp;gt;                                                          δ              →              0                                            (  1  +  δ  )  ln  ⁡      (    1    +          1      δ        )    =  ∞  .Conclusion      lim                                                      &amp;gt;                                                          δ              →              0                                                ∫    0    2              1      −      x              δ      +              x                  α                         d  x  =      {                            ∞                          for           α          ≥          1                                                  ∫            0            2                                              1              −              x                                      x                              α                                                         d          x                          for           α          &amp;lt;          1                        Conclusion      lim                                                      &amp;gt;                                                          δ              →              0                                                ∫    0    2              1      −      x              δ      +              x                  α                         d  x  =      {                            ∞                          for           α          ≥          1                                                  ∫            0            2                                              1              −              x                                      x                              α                                                         d          x                          for           α          &amp;lt;          1

For any α ∈ <mrow class="MJX-TeXAtom-ORD"> <mi mathvari

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