What is the limit as x approaches negative infinity of x + x 2 + 2 x ?

Question

What is the limit as x approaches negative infinity of       x    +                            x          2                +        2        x            ?

Hussut8tv · Accepted Answer

First, we&#039;ll try to plug in the value:            lim              x        →        -        ∞              x    +                            x          2                +        2        x              =    -    ∞    +                  ∞        -        ∞            We&#039;re already encountering a problem: it is simply not allowed to have       ∞    -    ∞  , it&#039;s like dividing by zero.We need to try something new.When I come across a limit like this, I use the following trick:            lim              x        →        -        ∞              x    +                            x          2                +        2        x                  =          lim              x        →        -        ∞              x    +                            x          2                +        2        x              ⋅                  x        -                                            x              2                        +            2            x                                      x        -                                            x              2                        +            2            x                              These are the same becaus the factor we&#039;re multiplying with is essentially 1.What are we doing? Because there is a formula that states:             (      a      -      b      )              (      a      +      b      )        =          a      2        -          b      2      In this case       a    =    x   and       b    =                            x          2                +        2        x            Let us try this formula:            lim              x        →        -        ∞                                      x          2                -                              (                                                            x                  2                                +                2                x                                      )                    2                            x        -                                            x              2                        +            2            x                                    =          lim              x        →        -        ∞                                      x          2                -                  x          2                -        2        x                    x        -                                            x              2                        +            2            x                                    =          lim              x        →        -        ∞                            -        2        x                    x        -                                            x              2                        +            2            x                              We&#039;re going to try another trick now. We&#039;r going to use this one, because we want to get the             x      2       out of the square root:            lim              x        →        -        ∞                            -        2        x                    x        -                                            x              2                                      (              1              +                              2                x                            )                                          Now, you might say that                               x          2                      =    x  , but you have to remember that x is a negative number. We&#039;re taking the positive square root, so                               x          2                      =    -    x   in this case.      =          lim              x        →        -        ∞                            -        2        x                    x        +        x                              1            +                          2              x                                                =          lim              x        →        -        ∞                            -        2        x                    x                  (          1          +                                    1              +                              2                x                                              )                    We can cancel the x:      =          lim              x        →        -        ∞                            -        2                    1        +                              1            +                          2              x                                          Now we can finally enter the value:      =    -          2              1        +                              1            +                          2                              -                ∞                                                        A number divided by infinity, is always 0:      =    -          2              1        +                              1            +            0                                =    -          2              1        +        1              =    -          2      2        =    -    1

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