# What is a mathematical limit?

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