Q. 43 - Calculus

Question

For each limit statement in Exercises $41-44$, use algebra to find $\delta$ or $N$ in terms of $\epsilon$ or $M$, according to the appropriate formal limit definition.

$\underset{x\to 1^{-}}{\lim }\frac{1}{1-x}=\infty$, find $\delta$ in terms of $M$.

Step-by-Step Solution

Answer

For $\underset{x\to 1^{-}}{\lim }\frac{1}{1-x}=\infty$, $\delta =\frac{1}{M}$.

Step 1 . Given information

$\underset{x\to 1^{-}}{\lim }\frac{1}{1-x}=\infty$.

Step 2 . From the limit expression,

$$\begin{gathered} f\left(x\right)=\frac{1}{1-x} \\ c=1 \\ L=\infty \end{gathered}$$

Now for $\delta >0$.

$$\begin{gathered} \left|x-1\right|<\delta \\ \left|1-x\right|<\delta \\ \frac{1}{\left|1-x\right|}<\delta \\ \left|\frac{1}{1-x}\right|<\delta \\ <M \end{gathered}$$

Hence, $\delta =\frac{1}{M}$.