How do you use the definition of continuity and the properties of limits to show that the function g ( x ) = - x 2 + 8 ⋅ x - 15 is continuous on the interval [3,5]?

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How do you use the definition of continuity and the properties of limits to show that the function             g              (        x        )              =                  -                  x          2                +        8        ⋅        x        -        15             is continuous on the interval [3,5]?

cdtortosadn · Accepted Answer

In order for g to be continuous on [3,5], the definition of continuous on a closed interval requires:For c in (3,5), we need             lim              x        →        c                    g              (        x        )              =          g              (        c        )            and we also need one-sided continuity at the endpoints:we need:             lim              x        →                  3          +                            g              (        x        )              =          g              (        3        )             and             lim              x        →                  5          -                            g              (        x        )              =          g              (        5        )            For c in (3,5), We&#039;ll use the properties of limits to evaluate the limit:            lim              x        →        c                    g              (        x        )              =          lim              x        →        c                            -                  x          2                +        8        x        -        15                  =                            lim                      x            →            c                                    (          -                      x            2                    +          8          x          -          15          )                          =                            lim                      x            →            c                                    (          -                      x            2                    )                +                  lim                      x            →            c                                    (          8          x          )                -                  lim                      x            →            c                                    (          15          )                          =                  -                  lim                      x            →            c                                    (                      x            2                    )                +        8                  lim                      x            →            c                                    (          x          )                -                  lim                      x            →            c                                    (          15          )                          =                  -                              (                          lim                              x                →                c                                                    (              x              )                        )                    2                +        8                  lim                      x            →            c                                    (          x          )                -                  lim                      x            →            c                                    (          15          )                          =                  -                              (            c            )                    2                +        8                  (          c          )                -                  (          15          )                          =          g              (        c        )            Use the one-sided versions of the limit properties at the endpoints.For c=3, replace all limits of the form             lim              x        →        c             with             lim              x        →                  3          +                    For c=5, replace all limits of the form             lim              x        →        c             with             lim              x        →                  5          -

How do you use the definition of continuity and the properties of limits to show that the function g(x) = sqrt(-x^2 + 8*x - 15) is continuous on the interval [3,5]?

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