# Real Analysis, Limit of a Function

• April 14th 2009, 07:18 PM
jstew
Real Analysis, Limit of a Function
Prove that $\lim_{x\to c}f(x) =L$ if and only if $\lim_{x\to 0}f(x+c) =L$.

I know that I'm to use the epsilon-delta definition of a limit, but for some reason I just can't see how to go about this.
• April 15th 2009, 08:36 AM
Plato
Quote:

Originally Posted by jstew
Prove that $\lim_{x\to c}f(x) =L$ if and only if $\lim_{x\to 0}f(x+c) =L$.

From the given we know, $\varepsilon > 0\, \Rightarrow \,\left( {\exists \delta > 0} \right)\left[ {0 < \left| {y - c} \right| < \delta \, \Rightarrow \,\left| {f(y) - L} \right| < \varepsilon } \right]$.

Now observe: $0<\left| x \right| < \delta \, \Rightarrow \,0,\left| {\left( {x + c} \right) - c} \right| < \delta \, \Rightarrow \,\varepsilon > 0\, \Rightarrow \,\left[ {\,\left| {f(x + c) - L} \right| < \varepsilon } \right]$