Intuitive Idea of Limits

An introduction to limits

Let us consider a point in a domain (defined by its existence). The concept of a limit of the function at point a in its domain is the value that a function approaches as the argument of the function approaches point a. In simpler terms, let us consider an independent variable x. The intuitive approach to limits explains the phenomenon that as x approaches a point a, the value of the function at that point approaches a real number or when the input approaches a particular value. Here, the variable used is usually x, and L denotes the limit of the function. The limit of a function does not necessarily give you its value at a given point in a domain. What it does is to describe the limit’s behaviour at that particular instance or point.

Formal definition and motivation of function limits

Imagine Ron Weasley’s car moving towards Hogwarts. The graph this car follows is represented by the equation of y-f(x), where the horizontal position of this car is represented by x, and the height at which it is flying in or the altitude is denoted by y.  As the car approaches Hogwarts, Harry Potter and Ron notice that the car’s altitude approaches L. But what does this mean? Let us exploit this example of Ron’s flying car to understand this scenario. Let’s say L is the point where they are supposed to park in Hogwarts, which, in this context, is the goal we set for Ron’s car, the variable. Our goal for this variable is that Ron needs to land the car within a 10m radius on the front porch. There are two factors that Ron’s flying car is following here.

1. It is moving towards Hogwarts horizontally.
2. Because it is also flying, it also adheres to a vertical point at every instance on the graph.

Hence, Ron needs to ensure at least one of the following two criteria are met for our goal to be accurate.

1. His car is within 10 metres of the goal set vertically.
2. His car is within 10 metres of the goal set horizontally.

Assuming x, the horizontal distance to Hogwarts is maintained, at any given point, the car will fulfil the horizontal criteria as it approaches L.

The concept of LIMIT

Let’s now consider a real-life example to understand the concept of a limit. Let’s say a friend hands you 12 inches of thread, and you are tasked with making the biggest rectangle you can with it. Let’s first denote the width by w and the length by l.  By this definition, 2w+2l=12 which means l=6—w. Hence, the area of this rectangle you will be creating, A=lw. = (6-w) w =6w-w². Using this formula, you now get to play around with different dimensions and will reach a point where you get the largest dimension in terms of area. Let’s call this the limit for now.

Width (w)	2	3	4	5	6
Area(A)	8	9	8	5	0

You soon realise the highest value in terms of area you can achieve at 3. Employing this example into the usage of limits, we can safely say that the A’s limit as X approaches the number 3 is 9. This holds true for any goal or limit we set for our variable.

Estimating a limit in numerical terms

Let’s consider it as our equation for understanding how to estimate a limit to concrete our understanding of the intuitive idea of limits. The way we read this equation is by considering the following legend. = limit of x till it reaches the real number two. X is our variable, and 2 is our limit. Let us revamp this formula to read as f(x)=(3x-2).We will create a table just like the one we used in the build-a-rectangle example. Now that we know how to read a basic limit equation and the variables involved, we should find the highest value around the limit, which, in this example, is 2.

x	1.9	1.99	1.999	2	2.001	2.01	2.1
f(x)	3.700	3.970	3.997	4.000	4.003	4.030	4.300

We understand from this table that x approaches 2, f(x) approaches 4.

Estimating a limit using a graph

Now that we know how to identify a limit in a simple equation, let’s see how to identify a limit. Although the equation looks different than the one we initially saw, the fundamental concept remains the same. X is the variable, f(x) is the limit. Let’s look at what the graph equation is supposed to represent. How did we arrive at this point on the graph, though? Let’s try to understand what the graph represents visually.

1. The graph runs through 2 for the value of +x at all instances on x on the graph.
2. The graph coincides at one point, which is 0=3 for all values of x approaching 3. This will hold true for ANY value for the limit as f(3)=0 has no value on the limit of x as it approaches 3.
3. We know the limit for f(x)=2  for all instances, except for when x=3, so we can safely assume that the limit, f(x), is 2.

 One-sided limits

One-sided limits are sequences that approach a value from either side at any given instance as opposed to both positive and negative spectrum number lines. The first equation we saw earlier was an example of a one-sided limit, which displays that the variable is approaching the limit of +2. This makes it a right-side limit. In order for this equation to be a left-sided limit, the equation will have to be written as where x is approaching the limit -2.

Two-sided limits

By definition, two-sided limits are one-sided limits for an equation that are the same. For example, if both approach the same number, say 2, the equation is a two-sided limit.

Non-existent limits

There are instances where limits fail to exist. Some of the scenarios where this might happen are:

1.     One-sided equations are approaching different limits. For example, it is a non-existent limit as |x|/x =1for all instances of x>0, and |x|/x=-1  for all the cases where x<0. In this case, no matter how close the limit approaches 0, there will always be a value in all the cases.
2.     The limit changes as the variable approaches a value.
3.     The limit oscillates between two fixed values as the variable approaches a value.

Conclusion

Limit Calculus is daunting to look at, and that is only because limits are defined and are limited by algebraic expressions. Limits are beneficial in understanding their applications in maximisation. A small example would be to employ our understanding of limits to identify the volume of an open swimming pool.