Infinite false limit verification via N−M method lim x → − ∞ x 3 + 6 2 x 2 − 5 = + ∞

Question

Infinite false limit verification via N−M method      lim          x      →      −      ∞                          x        3            +      6              2              x        2            −      5        =  +  ∞

Allison Pena · Accepted Answer

The error in your approach lies in the fact that from the inequalities                    (1)                              x          3                +        6        ⩽                                            x              3                        3                              and  2      x    2    −  5  ⩽  2      x    2  there is nothing that you can deduce about the quotient                               x          3                +        6                    2                  x          2                −        5            . However, from (1) and   2      x    2    −  5  ⩾      x    2   (which holds when   x  ∉      [    −          5        ,          5        ]  ), you can deduce that                    (2)                                                        x              3                        +            6                                2                          x              2                        −            5                          ⩽                  x          3                .            So, and since       lim          x      →      −      ∞                  x      3        =  −  ∞,       lim          x      →      −      ∞                                    x          3                +        6                    2                  x          2                −        5              =  −  ∞ too. On the other hand, it follows from (2) that  x  &amp;lt;  −      5      ⟹                                x          3                +        6                    2                  x          2                −        5              &amp;lt;  0  ,and so you cannot possibly have      lim          x      →      −      ∞                                    x          3                +        6                    2                  x          2                −        5              =  ∞  .

Infinite false limit verification via N−M method <munder> <mo movablelimits="true" form="pre

Answered question

Answer & Explanation

New Questions in Differential Calculus