One of the essential aspects of Calculus is Limits. Limits are useful for defining integrals, continuity, and also the derivatives. Let’s discuss the limit of a function:
If we take a function “f” which can be defined as some open interval and consists of numbers such as “a” or might be at “a” itself. In that case, you can write the limit of a function like:
lim x → af(x) = L, if given e > 0, there exists d > 0 such that 0 < |x – a| < d implies that |f(x) – L| < e. This means that the limit f(x) as “x” reaches “a” is “L.”
Limits and Continuity:
They are one of the most important topics as far as Calculus is concerned. Since the limit is already discussed, let’s take a look at what continuity is all about.
The topic of continuity is interesting and important too. A very easy way for testing continuity is to find out whether a pen can follow the graph of a function without you taking it off from a paper. While studying both precalculus and calculus, you need to understand the conceptual definition; however, moving ahead, a technical explanation is also necessary. With limits, the way for you to define continuity will be simple.
Continuity and Discontinuity:
As we already have understood that continuity can be defined as a pen following the graph’s function without lifting it up from a paper, here is what discontinuity is all about.
Discontinuity is of four types: Removal, Infinite, and Jump.
- Removal Discontinuity is expressed as: f(a) = lim x → af(x) f(a) = lim x → af(x)
- Infinite Discontinuity is expressed as:
lim x → 0− f (x) = lim x → 0 − xsin1x = 0 [Since -1 Similarly, lim x → 0 + f(x) = lim x → 0 + xsin1x = 0, [f(0) = 0]. Thus, lim x → 0−f (x) = lim x → 0 + f(x) = f(0).
- Jump Discontinuity is expressed as: lim x → a + f(x) ≠ lim x→a − f(x)