Is f ( x ) = x 2 - 9 x - 3 continuous at x=3?

Question

Is             f              (        x        )              =                            x          2                -        9                    x        -        3             continuous at x=3?

alienceenvedsf0 · Accepted Answer

f              (        x        )              =                            x          2                -        9                    x        -        3              is not continuous at x=3.In order for a function f(x) to be continuous at a given x-value a, the following condition must be satisfied:                              [          1          ]                            lim              x        →        a                    f              (        x        )              =          f              (        a        )            What this is saying is that, as x gets closer to a, f(x) should also get closer to f(a).For the given function f(x), the limit on the left-hand side of [1] will evaluate correctly. You&#039;ll end up with             lim              x        →        3                                      x          2                -        9                    x        -        3              =    6    .   However, the right-hand side of [1] presents a problem: what is f(x) when x=3? The answer is, it is not defined, because at that point, we have f(x) &quot;equal&quot; to 0/0:            f              (        3        )              =                            3          2                -        9                    3        -        3              =                  9        -        9            0        =          0      0      And this &quot;value&quot; of 0/0 is indeterminate.Thus, the function &quot;breaks&quot; at x=3, and so, because there is no f(3), it is not possible to say              lim              x        →        3                    f              (        x        )              =          f              (        3        )              .  . Thus, f(x) is not continuous at x=3.

Pader6u · Accepted Answer

Here&#039;s a tip to determine where some functions will be discontinuous. Any x-value that makes any denominator in the function equal to 0 is a point of discontinuity. So for the function above, x−3=0 when x=3, and so the function will be discontinuous at x=3.The only exception to this is if the function is piecewise defined at potential &quot;breaking&quot; points, like this:                    f        ⋆                    (        x        )              =          {                                                                                    x                  2                                -                9                                            x                -                3                                                                        ,                                            x            ≠            3                                                6                                              ,                                            x            =            3                              In this case, since f*(x) is defined at x=3, we have             lim              x        →        3                            f        ⋆                    (        x        )              =                  f        ⋆                    (        3        )              ,   so f*(x) is continuous at x=3 (and everywhere).

Is f(x)=(x^2-9)/(x-3) continuous at x=3?

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