Tsunoyukami
- 213
- 11
I'm working through some problems from Stewart's Calulus, 6ed. and am having some difficulty with certain limit proofs. In particular, there is no definition provided for limits of the form:
$$ \lim_{x \to - \infty} f(x) = L $$
One of the exercises is to come up with a formal definition for such a proof - which I did and used successfully a few times about a month ago when I was working through that section - but I want to make sure that my definition is really correct and not just lucky for those two problems.
I would suspect the definition of such a limit to be something like:
We say
$$ \lim_{x \to - \infty} f(x) = L $$
if for every ##\epsilon>0## there exists ##N## such that ##x < N \implies |f(x) - L| < \epsilon##.
Is this the correct precise definition of such a limit? I was unable to find an answer browsing google - most websites seem to provide the precise definition for "normal" limits and for limits at positive infinity but not at negative infinity.
$$ \lim_{x \to - \infty} f(x) = L $$
One of the exercises is to come up with a formal definition for such a proof - which I did and used successfully a few times about a month ago when I was working through that section - but I want to make sure that my definition is really correct and not just lucky for those two problems.
I would suspect the definition of such a limit to be something like:
We say
$$ \lim_{x \to - \infty} f(x) = L $$
if for every ##\epsilon>0## there exists ##N## such that ##x < N \implies |f(x) - L| < \epsilon##.
Is this the correct precise definition of such a limit? I was unable to find an answer browsing google - most websites seem to provide the precise definition for "normal" limits and for limits at positive infinity but not at negative infinity.
