One confusing, but fundamental, fact about a mathematical limit of a function f(x) as x approaches some number c is that the value of f(c) is technically irrelevant, though often useful (when the function is continuous at c).
For example, if f(x)=(x^2+2x-3)/(x-1) as in the example above, technically the value f(1) is undefined. However, lim_{x->1}f(x)=4 because the outputs of f(x) can be made as close to 4 as we want by taking x sufficiently close to, but not equal to, 1. For instance, if we want the value of f(x) to get within a distance 0.1 of 4, we can take x to be within a distance 0.1 of 1 (note that, for example, f(0.95)=3.95 and f(1.05)=4.05).
Why does this happen for this example? Because we can factor the top to get f(x)=((x-1)(x+3))/(x-1) and then cancel the x-1 factor to say f(x)=x+3 when x is NOT equal to 1. So the function f(x) has a graph that is a straight line with a slope of 1 and a y-intercept of 3, except that the point (1,4) is "missing" from the graph (the graph has a "hole" in it). In other words, f(x) is not continuous at x=1.
A continuous function, whose graph can be drawn without picking up your pencil, such as f(x)=x^2, can have its limit evaluated as x approaches any number c just by finding f(c).
An interesting example involving a trigonometric function to consider is lim_{x->0}(sin(x))/x. See if you can find this limit and prove that you are right.