Finding Limits from Graphs

We've seen how to compute limits using a table, by tracing along a function toward the point that we're interested in. Here, we'll see how to do the same thing using a graph, so that we don't have to go through the tedium of building a table.

Take a look at the graph below.

This graph is a bit contrived, but it'll give us a chance to look at a variety of cases all at once. Before we begin, take a good look at the graph and notice that there are several empty circles. These show where there are holes in the graph; in other words, this function is undefined at $(2,4)$. At $x=4$, there's a strange situation: the graph meets there at a hole, and then there's a solid dot at $(4,5)$, meaning that $f(4)=5$. Then at $x=5$, there's a break in the graph; just before $x=5$ the function is defined by the curve on the left, but at $5$ exactly, the function jumps to the lower curve, so $f(5)=1$.

Okay, once we're comfortable with this graph, we can try calculating some limits.

$\displaystyle\lim_{x \to 1} f(x)$

Let's start with an easy one. There's nothing strange happening at $x=1$, so as you approach from both sides, walking along the graph, the function values simply approach $f(1)$, which is $1$, so $\lim_{x \to 1} f(x) = 1$

$\displaystyle\lim_{x \to 2} f(x)$

This one is a bit more interesting, because of the hole in the graph at $x=2$, but it turns out that the limit isn't any harder. Remember, the limit never looks directly at the point we're interested in; it has no clue what's happening exactly at $x=2$, just what's going on nearby. Since the hole is only at the exact point where $x=2$, it doesn't affect the limit. As we trace along the graph, approaching $2$ from either side, the function values approach $4$, which is what the value of the function would be if that hole were filled in. Therefore, $\lim_{x \to 2} f(x) = 4$

$\displaystyle\lim_{x \to 4} f(x)$

This one is similar to the last one; there's a hole at $x=4$, but here we have the added complication of a single point defined at $(4,5)$. However, just as in the last case, the limit doesn't see exactly what's happening at $x=4$, so that weird point doesn't change anything, and $\lim_{x \to 4} f(x) = 4$

$\displaystyle\lim_{x \to 5} f(x)$

Finally we get to a more complicated example, and before we continue, we need to introduce a new term: one-sided limits.

One-Sided Limits

So far, whenever we've evaluated a limit, we've approached the point in question from both sides and looked for a trend, for the function values to approach some number. This is our first example of the two sides disagreeing. If you look at the graph, you should be able to see that as you approach $x=5$ from the left, the function values approach $2$, and as you approach from the right, the function values approach $1$. We can add a little bit of notation (a $+$ or $-$ sign) to indicate the direction from which we're approaching. We can write $\lim_{x \to 5^-} f(x) = 2$ and $\lim_{x \to 5^+} f(x) = 1$ to show what's going on here. Notice that the $+$ sign indicates that we're approaching from the right (using values slightly larger than $5$), and the $-$ sign is the opposite.

If we don't include a $+$ or $-$ sign in the limit notation, that implicitly refers to the two-sided limit.

In this case, since the one-sided limits do not agree, we say that the two-sided limit does not exist: $\lim_{x \to 5} f(x) \textrm{ DNE}$

In short, $\lim_{x \to c} f(x) = L \textrm{ iff (if and only if) } \lim_{x \to c^-} f(x) = L \textrm{ and } \lim_{x \to c^+} f(x) = L$