What is a mathematical limit?

1 Answer
Feb 12, 2015

One confusing, but fundamental, fact about a mathematical limit of a function f(x)f(x) as xx approaches some number cc is that the value of f(c)f(c) is technically irrelevant, though often useful (when the function is continuous at cc).

For example, if f(x)=(x^2+2x-3)/(x-1)f(x)=x2+2x3x1 as in the example above, technically the value f(1)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.