- #1
fogvajarash
- 127
- 0
Homework Statement
Show that ##\lim_{x \to a} f(x) = L## if and only if ##\lim_{x \to 0} f(x+a) = L##
Homework Equations
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The Attempt at a Solution
For the forward direction (ie ##1 \Rightarrow 2##), I tried to first assume that 1. holds true (ie ##\forall \epsilon>0, \exists \delta>0, \forall x \neq a, x \in A, |x-a|<\delta \Rightarrow |f(x)-L|<\epsilon##). Then, I let ##x+a=t##, and then by our choice of ##t## we will have ##|x|=|t-a|##, then if we choose ##\delta## such that ##|t-a|<\delta##, it follows ##|f(t)-L|<\epsilon##, so ##|f(x+a)-L|<\epsilon## proving statement 2.
However, I'm not quite sure if my argument is correct. The main concern I have is the chance that ##t \notin A##, so we couldn't apply the definition to this case. If the above can't be done, which would be a better way to solve the problem?
Thank you for your help.
However, I'm not quite sure if my argument is correct. The main concern I have is the chance that ##t \notin A##, so we couldn't apply the definition to this case. If the above can't be done, which would be a better way to solve the problem?
Thank you for your help.