Mathematics is tough. It is a subject that has a clear dichotomy between those who know and those who have goosebumps anytime they see an 'x' symbol. I definitely belonged to the latter, and never quite understood those selected few belonging to the former.
To be fair, I still do not quite grasp the wonderful world of Mathematics on its fullest. Friends (belonging to the I-love-Maths team) explained me about shapes, symmetry, elegance, etc. I get it. It is charming. But so are the other disciplines. I am not particularly more excited about Mathematics than I am about, say, Biology. But over time – and endless hour of persistence – I started to see how useful Mathematics can be. Don't get me wrong – I still struggle a lot with calculations, algebra, and anything related to finding the answers, remembering theorems, or proofs. Yet, I still appreciate the world of Mathematics because of their unique ways to see and describe the world.
A simple – yet very useful – example is that of the concept pertaining to limits. The concept of limit is essential to understand Calculus. Perhaps you are like me, and struggled the first time you heard about limits to understand what it actually meant. It took me few good years (yes, years, I am that slow) to actually think about limits in an intuitive way. Limits are the behaviour you would expect a function to have as an index changes. Got it?! Of course not. Let me elaborate a bit more with an example.
You have probably watch Youtube videos. Many have a title at the start of the video, and credits at the end. For this example, let's constrain our universe of potential videos to those with a title at the start and credits at the end. At the bottom of every video, you have a slide bar, which shows you how long the video is and how far along the video you are at any given point in time. Now, you know what to expect when the video is at the start (e.g., title) or at the end (credits). As you move the slide bar closer to the end of the video, you expect to reach the credits sooner than if you move the slide bar closer to the beginning. Correct?
This can be seen as an analogy to limits. The slide bar represents the value of a variable (let's call this variable a). The Youtube video is how the function that uses the value of the variable a (let's call it f(a) for the 'function of a'). This function f(a) has the following instruction: for any value of a, the function takes a and multiply it by itself (a*a). Note that the bigger the value of a gets, the bigger the value of the function f(a) gets. In this case, you expect that, if you take very large a, your function will also be very large! More formally, this means that as a tends to infinity ('as a gets larger'), then the function f(a) also tends to infinity (f(a) also gets larger). This, right here, is the idea of a limit.
In Mathematics, limits are useful because it enables a wide range of calculations. But for us, Mathematically-naive souls, limits are interesting and useful because they allow us to gain a very quick understanding of how a function, whatever it may be, will behave if we were to plug a very large (or very small) number in it. This gives us the first clue as to how the function may behave, and may be what we need to extract meaningful information for ourselves and our work.