How do you prove that the function f ( x ) = ( x + 2 x 3 ) 4 is continuous at a =-1?

Question

How do you prove that the function             f              (        x        )              =                  (        x        +        2                  x          3                )            4       is continuous at a =-1?

isscacabby17 · Accepted Answer

Take             lim              x        →        -                  1          +                            f              (        x        )              ,          lim              x        →        -                  1          -                            f              (        x        )            . If these limits are equal, we have continuity at x=−1.Explanation:Take the left and right hand limits of f(x) as       x    →    -    1  . If            lim              x        →        -                  1          +                            f              (        x        )              =          lim              x        →        -                  1          -                            f              (        x        )            , then f(x) is continuous at x=−1:            lim              x        →        -                  1          +                                    (        x        +        2                  x          3                )            4        =                  (        -        1        +        2                              (            -            1            )                    3                )            4        =                  (        -        3        )            4                  lim              x        →        -                  1          -                                    (        x        +        2                  x          3                )            4        =                  (        -        1        +        2                              (            -            1            )                    3                )            4        =                  (        -        3        )            4      Note that the direction from which we approached −1 did not change how the limits were evaluated, as this is a polynomial.These limits are equal; therefore, f(x) is continuous at x=−1.

How do you prove that the function f(x) = (x + 2x^3)^4 is continuous at a =-1?

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