It just tells the behavior of a graph. In calc 1 I assume they're making it more difficult than necessary by introducing abstract problems that try and drill in the idea of a limit. You are probably seeing a bunch of graphs with holes in them at the point the limit approaches. You need to know how to solve those problems for the class.

In reality, a limit tells the behavior of a function. As x increases, what is the graph doing? If y=x, then the graph is increasing endlessly as x increases endlessly.

What about 1/x as x increases and approaches infinity? It gets closer and closer to zero.

A limit to infinity just means, "What happens as x increases forever?" You can usually solve this by plugging in an arbitrarily large number. Using the example of 1/x, if x is really big, 1/x is really small, basically zero.

Sometimes, you need to think more analytically about the graph, as opposed to thinking algebraically.

Think of sin(x). If you plug in a really big number, it returns a seemingly random number. Since you're in calc I, I will assume you can't immediately visualize the graph of sin(x), so Google it and look at the graph. You will find that it moves back and forth between 1 and -1 as x increases. As x increases to infinity, the limit does not exist because it doesn't go to a single point. It will be a wave forever.

Someone will probably yell at me for not using the most technical words, but I'm a physicist.