lim ( x 2 + 1 − 2 ) / ( x − 1 )x approaches 1

Question

lim  (            x              2              +    1    −      2    )      /    (  x  −  1  )x approaches 1

jbacapzh · Accepted Answer

lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =Evaluating gives 0/0, indicating a common factor of x-1. So we need to manipulate this a little to exposethis, so we can cancel it and then try to evaluate the limit. Multiplying the top and bottom by the conjugate      lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      lim          x      →      0                                    x                      2                          +        1            −              2                    x      −      1        ∗                              x                      2                          +        1            +              2                                      x                      2                          +        2            +              2            Simplifying      lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      lim          x      →      0                          x                  2                    +      1      −      2              (      x      −      1      )      [                        x                      2                          +        1            +              2            ]            lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      lim          x      →      0                          x                  2                    −      1              (      x      −      1      )      [                        x                      2                          +        1            +              2            ]            lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      lim          x      →      0                  (      x      −      1      )      (      x      +      1      )              (      x      −      1      )      [                        x                      2                          +        1            +              2            ]      And there it is, canceling      lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      lim          x      →      0                  x      +      1                                x                      2                          +        1            +              2            Evaluating      lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =            1      +      1                      1        +        1            +              2                  lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      2                  2            +              2                  lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      2          2      +              2                  lim          x      →      1                                    x                      2                          +        1            −              2                    x      −      1        =      1          2

lim(sqrt(x^2 +1)-sqrt(2))/(x-1) x approaches 1

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Answer & Explanation

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