Question

Let

ƒ(x) = 𝑥1/(1−x) .

Make tables of values of ƒ at values of x that approach x = 1 from above and below. Does ƒ(x)

appear to have a limit as x approaches 1? If so, what is it? If not, why not?

Answer 1

Let $y=f(x)=\dfrac{x+1}{1-x}$

$$\def\arraystretch{1.5} \begin{array}{c:c} x & y \\ \hline 0.9 & 19 \\ 0.99 & 199 \\ 0.999 & 1999 \\ 0.9999 & 19999 \\ 1.0001 & -20001 \\ 1.001 & -2001 \\ 1.01 & -201 \\ 1.1 & -21 \\ \end{array}$$

$$\lim\limits_{x\to1^-}f(x)=\infin$$

$$\lim\limits_{x\to1^+}f(x)=-\infin$$

Since $\lim\limits_{x\to1^-}f(x)\not=\lim\limits_{x\to1^+}f(x),$ then $\lim\limits_{x\to1}f(x)$ does not exist.