اگر $f(x)=\left\{ \begin{matrix} \frac{x-\sqrt{x}}{۱-{{x}^{۲}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \gt ۱ \\ a\left[ -x \right]+\left[ -۳x \right]\,\,\,\,\,\,x \lt ۱ \\ b+۱\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=۱ \\ \end{matrix} \right.$ در $x=۱$ پیوسته باشد، $\underset{x\to {{(-۱)}^{-}}}{\mathop{\lim }}\,f(x)$ کدام است؟ ($\left[ \, \right]$ نماد جزء صحیح است.)
$f(1)=b+1$ $\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,(a\left[ -x \right]+\left[ -3x \right])=a\left[ -{{(1)}^{-}} \right]+\left[ -{{(3)}^{-}} \right]=-a-3$ $\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,\frac{x-\sqrt{x}}{1-{{x}^{2}}}=\underset{x\to 1}{\mathop{\lim }}\,\frac{1-\frac{1}{2\sqrt{x}}}{-2x}=\frac{\frac{1}{2}}{-2}=-\frac{1}{4}$ $\left\{ \begin{matrix} b+1=-\frac{1}{4}\Rightarrow b=-\frac{5}{4} \\ -a-3=-\frac{1}{4}\Rightarrow a=-\frac{11}{4} \\ \end{matrix} \right.$ $\underset{x\to {{(-1)}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to {{(-1)}^{-}}}{\mathop{\lim }}\,(-\frac{11}{4})\left[ -x \right]+\left[ -3x \right])=-\frac{11}{4}\left[ -{{(-1)}^{-}} \right]+\left[ -3{{(-1)}^{-}} \right]=\frac{-11}{4}\times 1+3=\frac{1}{4}$