# How do you use a graph to determine limits?

The limit of a function $f \left(x\right)$ at a given point $x = a$ is, essentially, the value one would expect the function $f \left(x\right)$ to take on at $x = a$ if one were going solely by the graph. For example, if given a graph which resembles the function $f \left(x\right) = x - 1$, one might expect the function to take on the value $f \left(x\right) = 0$ at $x = 1$. However, the function $f \left(x\right) = {\left(x - 1\right)}^{2} / \left(x - 1\right)$ would also be graphed like $f \left(x\right) = x - 1$, but would be undefined at $x = 1$.
In the case listed above, one would analyze the situation by examining the function's behavior in the graph for $x$-values slightly above and slightly below the desired point. For this case, suppose one examines the graph at the points $x = 0 , x = 0.5 , x = 0.75 , x = 1.25 , x = 1.5 , x = 2$. Doing this, we determine that as $x \to 1$ from both the right and the left, $f \left(x\right) \to 0$. Thus, the two-sided limit of the function $f \left(x\right) = {\left(x - 1\right)}^{2} / \left(x - 1\right)$ at $x = 1$ is 0, though $f \left(1\right)$ itself is undefined (as it takes on the form $\frac{0}{0}$)