Definition of a Limit: Subtle Differences

[FredericChopin]

Here is the definition of the limit of f(x) is equal to L as x approaches a:

"For every positive real number ϵ > 0 there exists a positive real number δ > 0 so that whenever 0 < |x − a| < δ, we have |f(x) − L| < ϵ."

But what is the difference if I use this definition?

"For every positive real number δ > 0 there exists a positive real number ϵ > 0 so that whenever 0 < |x − a| < δ, we have |f(x) − L| < ϵ."

 

[Mark44]

Here is the definition of the limit of f(x) is equal to L as x approaches a:

"For every positive real number ϵ > 0 there exists a positive real number δ > 0 so that whenever 0 < |x − a| < δ, we have |f(x) − L| < ϵ."

But what is the difference if I use this definition?

"For every positive real number δ > 0 there exists a positive real number ϵ > 0 so that whenever 0 < |x − a| < δ, we have |f(x) − L| < ϵ."

It's a big difference. In the first definition (the real one), it's saying that f(x) can be made arbitrarily close to L, the purported limit. No matter how close someone else requires, you can find an open interval around a so that for any x in that interval, f(x) is as close as required to L.

In your definition, you're saying that for x arbitrarily close to a, a number ϵ exists so that f(x) is within that distance of L. This doesn't guarantee that f(x) is actually close to L - only that the two are within some distance apart.

 

[Fredrik]

Consider the function $f:\mathbb R\to\mathbb R$ defined by $f(x)=0$ for all $x\in\mathbb R$. The second definition would make 55 a limit of $f(x)$ as $x\to 0$, because for all $\delta>0$, we have
$$0<|x|<\delta\ \Rightarrow\ |f(x)-55|<100.$$ In fact, there's no real number that isn't a limit of $f(x)$ as $x\to 0$ if you use this definition.