vertciel
- 62
- 0
Hello there,
I would like to learn how I can use the formal definition of a limit to prove that a limit does not exist. Unfortunately, my textbook (by Salas) does not offer any worked examples involving the following type of limit so I am not sure what to do. I write below that delta = 1 would seem to work because f(x) = 1/x increases without bounds on (0,1].
Thank you for your help.
---
\begin{align}<br /> & \text{Prove that }\underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}\text{ does not exist}\text{.} <br /> <br /> \end{align}
>\begin{align}<br /> & \text{I know that I must negate the limit definition, as such:} \\ <br /> & \forall \text{L,}\exists \varepsilon \text{0 st }\delta \text{0, }\left| x-c \right|<\delta \Rightarrow \left| f(x)-L \right|\ge \varepsilon \\ <br /> & \text{Also, I believe that if I take }\delta =1,\text{ this value will help me with the above}\text{.} \\ <br /> & \text{However, how would I go about doing this? } <br /> \end{align}
I would like to learn how I can use the formal definition of a limit to prove that a limit does not exist. Unfortunately, my textbook (by Salas) does not offer any worked examples involving the following type of limit so I am not sure what to do. I write below that delta = 1 would seem to work because f(x) = 1/x increases without bounds on (0,1].
Thank you for your help.
---
Homework Statement
\begin{align}<br /> & \text{Prove that }\underset{x\to 0}{\mathop{\lim }}\,\frac{1}{x}\text{ does not exist}\text{.} <br /> <br /> \end{align}
Homework Equations
The Attempt at a Solution
>\begin{align}<br /> & \text{I know that I must negate the limit definition, as such:} \\ <br /> & \forall \text{L,}\exists \varepsilon \text{0 st }\delta \text{0, }\left| x-c \right|<\delta \Rightarrow \left| f(x)-L \right|\ge \varepsilon \\ <br /> & \text{Also, I believe that if I take }\delta =1,\text{ this value will help me with the above}\text{.} \\ <br /> & \text{However, how would I go about doing this? } <br /> \end{align}