Finding Limits on a Graph
Most people mess this up because they're looking at the wrong part of the line. I spent three semesters grading calculus exams and the same mistake showed up in about 60% of submissions every time. Here's how to actually do it without second-guessing yourself. The graph is just a picture of the function. That's all you need to remember. A limit asks what y-value the function is approaching as x gets closer and closer to some number. Not what the function equals at that number. What it's heading toward. Start by identifying the x-value you're interested in. Let's call it a. Then trace the curve from both the left side and the right side as you approach a. If both sides meet at the same height, that's your limit. Simple enough in theory.
Here's where it gets messy in practice. I had a student once who couldn't figure out why his answer was wrong on a piecewise function with a removable discontinuity. The graph had a hole at x = 2, but the curve approached y = 5 from both directions. He wrote 5 as the limit and marked it wrong because the function value at x = 2 was actually 3. He kept insisting the answer had to be 3. I told him to cover the hole with his finger and look at where the lines were pointing. That usually clicks for people. One thing most textbooks don't emphasize enough: a limit doesn't care about the point itself. You can have a solid dot, a hole, an arrow, whatever. The limit only cares about the neighborhood around the point. Think of it like approaching a parking space. You don't need to be in the spot to know which spot it is. Let me walk through a concrete example. Say you have a graph where as x approaches 4 from the left, the y-values climb steadily toward 7. As x approaches 4 from the right, the curve drops down and also heads toward 7. The limit is 7. Now imagine the right side instead heads toward 2. Left approaches 7, right approaches 2. The limit doesn't exist. You'd write DNE or say the two-sided limit fails.
I once worked with a dataset where the function had an asymptote at x = -1. From the left, values shot up to positive infinity. From the right, they dropped to negative infinity. That's an infinite limit, but it's still a limit in the sense that the behavior is predictable. Just not a finite one. Students often confuse "limit doesn't exist" with "infinite limit." They're related but not identical. An infinite limit describes a specific kind of non-existence. Another counter-intuitive case that trips people up constantly: functions with oscillating behavior near a point. Take sin(1/x) as x approaches 0. The graph writhes faster and faster, never settling on any single value. The limit truly doesn't exist. There's no workaround. It's not that the graph is unclear. It's that the function is genuinely doing too many things at once. Vertical asymptotes are another common source of errors. When I see a graph with a vertical asymptote at x = c, I immediately check which direction each branch is heading. Sometimes both go to positive infinity, sometimes one goes positive and one goes negative. Either way, the two-sided limit doesn't exist, but noting the one-sided behavior is still useful for understanding the function.
Get the Full Details
Here's a practical tip that saves time during exams. When you're given a graph and asked for a limit, draw two arrows. One coming from the left, one from the right, both pointing at the x-value in question. Follow each arrow along the curve to see where it's headed on the y-axis. If they converge, you're done. If they diverge, state which values each side approaches separately. I've also seen people try to read the exact y-value by eye from a printed graph and get annoyed when their answer doesn't match the key. Graphs in textbooks are illustrations, not precision instruments. If the grid lines are spaced one unit apart and the curve passes somewhere between 4 and 5, you report what you can read. Sometimes the answer is meant to be an integer and the graph is just schematic. Don't overthink the pixel placement. The real pitfall I notice repeatedly is when students assume continuity without checking. Just because a graph looks connected doesn't mean there isn't a tiny break or a hole you can't easily see at that resolution. Zoom in if you're working digitally, or look for open circles on paper. An open circle means the point is excluded from the function, which is irrelevant for the limit but vital if the question also asks for the function value.
If you're using technology to verify your work, Desmos handles limit visualization decently. You can zoom in indefinitely on any point and watch the curve behavior change. WolframAlpha will tell you the limit, but it won't always explain why. That's on you to figure out by looking at the graph. Bottom line: limits on graphs are about direction, not destination. Watch where the curve is going, not where it lands. Check both sides. Ignore the actual point value unless asked. If the two sides disagree, the limit doesn't exist. That's about it.
