How to Graph Left-Hand and Right-Hand Limits

Hey there, math explorers! 🌟 Today, we’re going to jump into the awesome topic of limits. We will focus on something called one-sided limits!

These ideas help us understand how functions behave when they get close to a certain point. Are you ready to uncover the secrets of limits? Let’s get started!

What Are One-Sided Limits?

One-sided limits are like looking at a function as it moves toward a specific value, but only from one side. Think of them as secret agents:

1. Left-Hand Limit: This tells us the value the function gets close to as we approach from the left side. We write it like this: $\lim_{x \to c^-} f(x)$ The $c^-$ means we’re moving in from the left.

2. Right-Hand Limit: This one tells us the value the function gets close to as we come from the right side. We write it like this: $\lim_{x \to c^+} f(x)$ The $c^+$ means we’re moving in from the right.

Why Graph One-Sided Limits?

Graphing one-sided limits is not just fun—it's also super helpful! It lets us see how functions behave near specific points. By doing this, we can figure out if the function has a limit at that point and if the limits from both sides are the same. If they are, then the two-sided limit exists!

How to Graph Left-Hand and Right-Hand Limits

Steps to Graph One-Sided Limits

1. Choose a Function and a Point: Let’s pick a function $f(x)$ and see what happens as $x$ gets closer to $c$.

2. Draw the Graph: First, plot the whole graph of $f(x)$ on a coordinate plane.

3. Find the Point $c$: Locate the vertical line $x = c$ on your graph. This is the point we’ll focus on!

4. Look at the Left-Hand Limit:

   • Check the values of $f(x)$ as $x$ approaches $c$ from the left (values less than $c$).
   • Find the y-value that $f(x)$ gets close to when you’re near $c$ from the left side.

5. Look at the Right-Hand Limit:

   • Now check the values of $f(x)$ as $x$ approaches $c$ from the right (values greater than $c$).
   • Find the y-value that $f(x)$ gets close to when you’re near $c$ from the right side.

6. Mark the Limits on the Graph: Use open and closed circles to show whether the limits include that point:

   • An open circle means the function value isn’t defined at $c$.
   • A closed circle means the function value exists at $c$.

Example! 🎉

Let’s look at this function:

\begin{cases} 2x & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ x^2 & \text{if } x > 1 \end{cases} $$ - For the **left-hand limit** as $x$ approaches 1: $$ \lim_{x \to 1^-} f(x) = 2(1) = 2 $$ - For the **right-hand limit** as $x$ approaches 1: $$ \lim_{x \to 1^+} f(x) = 1^2 = 1 $$ ### Conclusion The left-hand limit is 2, and the right-hand limit is 1. Since they’re not equal, it means that the two-sided limit does not exist at $x=1$. Isn’t that interesting? Keep practicing, and you’ll become a limit master in no time! 🧙‍♂️✨