When you start looking at limits, especially through graphs, it feels a bit like being a detective. You're trying to figure out where a function is going.

So, how can you find out what values a function is approaching on a graph? Here are some simple tips based on what I’ve learned:

What Are Limits?

A limit is the value a function gets closer to as the input (or $x$-value) gets near a certain point. It's important to know that the limit isn’t always the same as the function's value at that point. Instead, it’s about how the function behaves around it.

Easy Steps to Look at Limits Using Graphs

1. Draw the Graph: Start by making the function's graph or using one that you've been given. If you have a graphing calculator or software, these tools can show the function clearly.

2. Choose the Point You Want to Check: Pick the $x$-value that you want to look at. This could be a spot where the function acts strangely, like changing direction or having a hole or a vertical line.

3. Follow the Graph: Watch how the graph behaves as $x$ gets close to the point you picked. Look at the values to the left ($x \to a^-$) and to the right ($x \to a^+$) of that point $a$.

Checking the Left and Right Limits

• Left-Hand Limit ($x \to a^-$): As you get closer from the left (values smaller than $a$), see if the $y$-values settle down to a certain number.

• Right-Hand Limit ($x \to a^+$): Now, do the same thing from the right (values larger than $a$) and check if the $y$-values get close to the same number as from the left.

Understanding the Behavior

• Both Sides Agree: If the left-hand limit and right-hand limit are the same (like both are approaching 2), then the limit as $x$ approaches $a$ exists and is that number. We write this as $\lim_{x \to a} f(x) = 2$.

• Different Values: If the two sides give different numbers (like the left goes to 2 and the right goes to 3), then the overall limit does not exist. We can note this as $\lim_{x \to a} f(x) \text{ does not exist}$.

Special Things to Look For

• Holes: If there’s a hole in the graph at point $a$, look for the limit around that hole. This often happens in functions where you can remove the issue.

• Vertical Asymptotes: If the graph goes up or down forever as you get close to $a$, that shows a vertical asymptote. In these cases, it's good to see if the limit goes to $\infty$ or $-\infty$.

Keep Practicing

Just like anything else in math, practice is key! The more different functions and their graphs you work on, the better you'll get at spotting limits quickly. Try different functions and check your understanding by comparing what you see with your calculations to improve your skills.

So, when you’re checking limits using graphs, just remember: it's all about how the graph acts as it gets closer to that important point!