Limits

A limit is a value of a function as it approaches a certain point. To put it simply, let's say we have a sequence of numbers: 1, 2, 3, 4, x, 6, 7, 8, 9. We'll let x be some undefined number; Nobody knows what it is because, well... it's not defined. But by looking at the sequence, we know it's probably 5. Well, that's kinda what limits are.

The image below is the graph of:

If you look at it, it's just like any other parabola... until you plug in x = 0. This will cause the denominator to become zero, and we know you can't divide by zero. Hence the hole.

By eyeballing it, it's clear that the point looks to be (0,2), but that's not its actual value. That is the 'limit of the function as it approaches x equals zero'.

Let's take a look at the following graph:

This function does not have a limit as it approaches x equals zero. The limit is simply undefined. Why? For a limit to be deemed valid, it must end up at the same number whether you approach it from the negative or positive sides.

Limits aren't only used to calculate the value of a non-continuous function. You can also use them to calculate the value of a function as it approaches infinity. Take a look at this graph:

As our function gets closer to infinity, the value gets closer to zero. From this observation, we can conclude that 'the limit of this function as it approaches infinity is zero'. In this case, the x-axis is called an asymptote as the curve keeps getting closer to the x-axis, but they never truly touch.

We can also use derivatives to compute limits with L'Hôpital's rule. L'Hôpital's rule state that:

I can't think of anything else I should add, so I'll end it here. I hope this post is simple yet clear enough to make you understand limits. If you have any feedback, feel free to post it in the comments section. Good luck with your Calculus classes!