Understanding Limits on Graphs- Practice Questions and Solutions
What Limits Actually Are (And Why Graphs Make Them Easier)
A limit describes what happens to a function as you get arbitrarily close to a specific x-value. Not what happens at the point—right next to it.
That's the part most textbooks gloss over. They say "approaches" like it's poetic. It's not. It's mathematical precision. You're asking: what y-value is the function hunting as x gets closer to a?
Why Graphs Beat Equations for Learning This
Because you can see the behavior. You spot holes, jumps, and curves. Your brain processes visual patterns faster than algebraic manipulation. When you see a graph approaching a point from both sides, you're doing limit analysis—without realizing it.
Let's build the skill properly.
Reading Limits From Graphs: The Core Method
Here's the procedure that works every time:
- Pick an x-value slightly less than your target (approach from left)
- Pick an x-value slightly greater than your target (approach from right)
- Read the corresponding y-values
- Ask: do they agree?
If both sides agree, the limit exists and equals that y-value. If they disagree, the limit does not exist (DNE).
That's it. Everything else is just context.
One-Sided vs Two-Sided Limits
A two-sided limit requires agreement from both directions. An one-sided limit looks at only one side.
Notation:
- Left-hand limit: lim(x→a⁻) f(x)
- Right-hand limit: lim(x→a⁺) f(x)
- Two-sided limit: lim(x→a) f(x)
The two-sided limit exists only when both one-sided limits exist AND agree.
Practice Questions and Solutions
Question 1
Use the graph below to find lim(x→2) f(x)
Solution: Trace your finger toward x=2 from the left. The y-value approaches 3. Now trace from the right. Still approaches 3. Answer: 3
Question 2
The graph has a jump at x=5. From the left, f(x)→4. From the right, f(x)→2. What is lim(x→5) f(x)?
Solution: The one-sided limits disagree (4 ≠ 2). The two-sided limit does not exist. This is a jump discontinuity.
Question 3
What is lim(x→0) f(x) if f(x) = sin(1/x) plotted near the origin?
Solution: This function oscillates infinitely between -1 and 1 as x→0. It never settles. The limit DNE. No matter how close you get, you can't pin down a single y-value.
Question 4
A function has a hole at x=3 with the point (3,7) missing, but the graph approaches y=7 from both sides. Find lim(x→3) f(x).
Solution: The limit is 7. The hole doesn't matter. Limits don't care what happens at the point—only what happens around it. The function never reaches 7, but it gets arbitrarily close.
Question 5
Find lim(x→∞) for the function that levels off at y=2 as x increases.
Solution: Horizontal asymptotes describe limits at infinity. As x grows without bound, if f(x) approaches 2, then lim(x→∞) f(x) = 2.
Common Graph Features That Affect Limits
| Feature | What It Means for Limits |
|---|---|
| Hole (point discontinuity) | Limit can still exist—function just doesn't reach it |
| Jump (step discontinuity) | One-sided limits differ—two-sided limit DNE |
| Vertical asymptote | Function blows up—limit DNE (or is ±∞) |
| Horizontal asymptote | Describes behavior at infinity—limit exists |
Getting Started: Your Limit-Reading Checklist
When you see a limit problem on a graph:
- Identify the x-value you're approaching
- Sketch light vertical lines at x = a - 0.1 and x = a + 0.1
- Read the y-values where your graph intersects these lines
- Compare—do they match?
- State your answer with the correct notation
Practice this with 10 graphs and it'll be automatic. You're not memorizing rules—you're training your eye to see mathematical behavior.
Quick Reference: Limit Notation
- lim(x→a) f(x) = L means: as x gets close to a, f(x) gets close to L
- lim(x→a⁻) means: x approaches a from below only
- lim(x→a⁺) means: x approaches a from above only
- lim(x→∞) means: x grows without bound
Keep this notation clean. Professors deduct points for sloppy notation even when your answer is right.