![non-zero over zero limits non-zero over zero limits](https://img.youtube.com/vi/PDReqvXfkBA/0.jpg)
In these terms, the error (\varepsilon) in the measurement of the value at the limit can be made as small as desired by reducing the distance (\delta) to the limit point. The letters \varepsilon and \delta can be understood as ” error ” and “distance,” and in fact Cauchy used \epsilon as an abbreviation for “error” in some of his work.
![non-zero over zero limits non-zero over zero limits](https://i.ytimg.com/vi/q6APL9H6bV0/maxresdefault.jpg)
Note that the limit of the denominator exists and is non-zero.
![non-zero over zero limits non-zero over zero limits](https://media.cheggcdn.com/media/852/852438cb-b2e8-42d7-98a1-939c3a4423d7/phpprtJCE.png)
Therefore, the limit of this function at infinity exists. It seems almost obvious that we should expect limit(x,x infinity) infinity. Limit of a Function at Infinity: For an arbitrarily small \epsilon, there always exists a large enough number N such that when x approaches N, \left | f(x)-L \right | < \varepsilon. This means that we’ll have a numerator that is getting closer and closer to a non-zero and positive constant divided by an increasingly smaller positive number and so the result should be an increasingly larger positive number. Note that the value of the limit does not depend on the value of f(p), nor even that p be in the domain of f.