# Precise Definition of a Limit at Negative Infinity

1. Aug 24, 2014

### Tsunoyukami

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.

2. Aug 24, 2014

### micromass

Staff Emeritus
Sounds good.

3. Aug 24, 2014

### Tsunoyukami

Thank you for your prompt reply!

4. Aug 24, 2014

### BvU

Would sound even better if you had written $x > N$, but I'm pretty sure the < was just a typo...

5. Aug 24, 2014

### vela

Staff Emeritus
The OP wrote it correctly because the limit is for $x$ going to $-\infty$.

6. Aug 24, 2014

### BvU

My apologies. Couldn't imagine such a quirk; proves how rusty one can get with old age...

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