f(x)=e^{-1/x2} if x is not =0 =0 if x=0 then f is continuous. how it can be proved.
Follow Math Help Forum on Facebook and Google+
Originally Posted by saravananbs f(x)=e^{-1/x2} if x is not =0 =0 if x=0 then f is continuous. how it can be proved. See if $\displaystyle \displaystyle \begin{align*} \lim_{x \to 0^-}e^{-\frac{1}{x^2}} = \lim_{x \to 0^+}e^{-\frac{1}{x^2}} = 0 \end{align*}$.
Originally Posted by saravananbs f(x)=e^{-1/x2} if x is not =0 =0 if x=0 then f is continuous. how it can be proved. We need to show that $\displaystyle \lim_{x\to0}f(x) = f(0).$ $\displaystyle \lim_{x\to0}f(x)$ $\displaystyle =\lim_{x\to0}e^{-1/x^2}$ Now evaluate the limit and show that it equals $\displaystyle f(0).$
yes at x=0, it is continuous but for other points , how its derived. is it like limit x-> a+h f(x) = limit x-> a-h f(x) =f(a)?
Last edited by saravananbs; Jun 13th 2012 at 08:45 AM.
Originally Posted by saravananbs yes at x=0, it is continuous but for other points , how its derived. One way is to say that $\displaystyle e^{-1/x^2}$ is continuous as the composition of continuous functions.
Originally Posted by emakarov One way is to say that $\displaystyle e^{-1/x^2}$ is continuous as the composition of continuous functions. i think three function are involved , e^x ,1/x, x^2 all are continuous. is it right.
All are continuous where x =/= 0.
Yes, all are continuous for x ≠ 0.
View Tag Cloud