Question

What articles, if any, should I study to fully understand a limit. I start Calculus 1 in a few weeks. Thanks

Answer 1

A limit is just what happens as a graph approaches a certain value. Let's compare two similar graphs to understand why the concept of a limit is necessary.

Look at the graph of y=(4x^2+x)/x and y=4x+1

What's the difference between these two graphs? They are almost entirely identical, the difference is that the second one has the x divided out. But notice that they have different domains because in the first one you can't have a value of x=0 because then you would be dividing by 0 which isn't allowed.

So because the domain of the first graph is all numbers except 0 and the second one is all numbers, there's a hole in that first graph only at the point at x=0. So what happens as we approach x=0? This is where the limit comes in.

\[\lim_{x \rightarrow 0} (4x^2+x)/x \]
So to find out, we just divide out the x's which is dividing by 0, and weren't allowed to do in algebra. But we're in big boy calculus now, and we can divide by 0 and go to infinity now! So we get:

\[\lim_{x \rightarrow 0} (4x+1) \]
Now when we plug in x=0 to find the limit we get 1, even though there's a hole there, we see that the function approaches 0 from both sides.

That's really it to them, everything else is just kind of more of this kind of thinking. What happens as the function approaches a value. For continuous functions the limit is equal to the value of the function at that value, if that doesn't confuse you too much.

Answer 2

That is true because lim x→0 (4x 2 +x)/x is very critical in calculus. knowing that would help you with most problems.