Concept of Limits

In calculus, a limit describes the value that a function approaches as the input (or $x$-value) gets closer and closer to a specific number. Limits are the foundational concept for both derivatives and integrals.

What is a Limit?

The notation for a limit is: $\lim_{x \to c} f(x) = L$ This is read as: "The limit of $f(x)$ as $x$ approaches $c$ is $L$." It means that as you pick $x$-values closer and closer to $c$, the corresponding $y$-values of the function get closer and closer to $L$.

Notice that we care about what happens near $c$, not necessarily what happens exactly at $c$. The function might not even be defined at $x = c$.

One-Sided Limits

Sometimes, a function approaches different values depending on which direction you are coming from. We use one-sided limits to describe this:

• Left-hand limit: $\lim_{x \to c^-} f(x)$ describes the value the function approaches as $x$ gets closer to $c$ from values strictly less than $c$.
• Right-hand limit: $\lim_{x \to c^+} f(x)$ describes the value the function approaches as $x$ gets closer to $c$ from values strictly greater than $c$.

When Does a Limit Exist?

For a general (two-sided) limit to exist, the function must approach the exact same value from both the left and the right.

$\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = L \text{ and } \lim_{x \to c^+} f(x) = L$

If the left-hand limit and the right-hand limit are different, we say that the limit does not exist (DNE).

Estimating Limits Using a Table

Let's estimate the limit: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$

If we try to plug in $x = 2$, we get $\frac{0}{0}$, which is undefined. However, we can use a table of values to see what the function approaches as $x$ gets close to 2.

Approaching from the left ($x \to 2^-$):

• $x = 1.9 \implies f(1.9) = 3.9$
• $x = 1.99 \implies f(1.99) = 3.99$

Approaching from the right ($x \to 2^+$):

• $x = 2.1 \implies f(2.1) = 4.1$
• $x = 2.01 \implies f(2.01) = 4.01$

From the table, both the left and right sides are approaching $4$. Therefore: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$

Limits from a Graph

When looking at a graph to find $\lim_{x \to c} f(x)$:

1. Trace the graph with your finger from the left towards $x = c$. The $y$-value you hit is the left-hand limit.
2. Trace the graph from the right towards $x = c$. The $y$-value you hit is the right-hand limit.
3. If your fingers meet at the same $y$-value, the limit exists and equals that value. If your fingers end up at different heights (a jump in the graph), the limit does not exist.