which of the following functions ( f ) has a removable discontinuity at ( a )? if the discontinuity is…

which of the following functions ( f ) has a removable discontinuity at ( a )? if the discontinuity is removable, find a function ( g ) that agrees with ( f ) for ( x\neq a ) and is continuous at ( a ). (if an answer does not exist, enter none.)\n(a) ( f(x)=\frac{x^{3}-1}{x - 1},a = 1)\nthe discontinuity is removable.\nthe discontinuity is not removable.\n( g(x)=)\n(b) ( f(x)=\frac{x^{3}-x^{2}-6x}{x - 3},a = 3)\nthe discontinuity is removable.\nthe discontinuity is not removable.\n( g(x)=)\n(c) ( f(x)=lfloorsin x\rfloor,a=pi ) (recall that ( lfloor h(x)\rfloor ) means the largest integer that is less than or equal to ( h(x) ).)\nthe discontinuity is removable.\nthe discontinuity is not removable.\n( g(x)=)

which of the following functions ( f ) has a removable discontinuity at ( a )? if the discontinuity is removable, find a function ( g ) that agrees with ( f ) for ( x\neq a ) and is continuous at ( a ). (if an answer does not exist, enter none.)\n(a) ( f(x)=\frac{x^{3}-1}{x - 1},a = 1)\nthe discontinuity is removable.\nthe discontinuity is not removable.\n( g(x)=)\n(b) ( f(x)=\frac{x^{3}-x^{2}-6x}{x - 3},a = 3)\nthe discontinuity is removable.\nthe discontinuity is not removable.\n( g(x)=)\n(c) ( f(x)=lfloorsin x\rfloor,a=pi ) (recall that ( lfloor h(x)\rfloor ) means the largest integer that is less than or equal to ( h(x) ).)\nthe discontinuity is removable.\nthe discontinuity is not removable.\n( g(x)=)

Answer

Explanation:

Step1: Analyze function (a)

Simplify $f(x)=\frac{x^{2}-1}{x - 1}=\frac{(x - 1)(x + 1)}{x - 1}=x + 1$ for $x\neq1$. The discontinuity at $x = 1$ is removable. The new function $g(x)=x + 1$ is continuous at $x = 1$.

Step2: Analyze function (b)

Factor $f(x)=\frac{x^{3}-x^{2}-6x}{x - 3}=\frac{x(x^{2}-x - 6)}{x - 3}=\frac{x(x - 3)(x+2)}{x - 3}=x(x + 2)$ for $x\neq3$. The discontinuity at $x = 3$ is removable. The new function $g(x)=x(x + 2)$ is continuous at $x = 3$.

Step3: Analyze function (c)

The function $y=\lfloor\sin x\rfloor$ has a non - removable discontinuity at $x=\pi$ because the left - hand limit $\lim_{x\to\pi^{-}}\lfloor\sin x\rfloor = 0$ and the right - hand limit $\lim_{x\to\pi^{+}}\lfloor\sin x\rfloor=- 1$.

Answer:

(a) The discontinuity is removable. $g(x)=x + 1$ (b) The discontinuity is removable. $g(x)=x^{2}+2x$ (c) The discontinuity is not removable. $g(x)=$ONE