Estimating Limit Values from Graphs
Why this matters
Imagine you're driving from Dallas to visit a friend in Austin for a big college football game. Your GPS says you'll arrive at their house at 3:00 PM. As you get closer, you can see the house, but you hit a sudden roadblock right on their street! You have to park a block away and walk.
Did you arrive at the house? Not exactly. But was the intended destination still your friend's house? Absolutely. The entire trip was planned around that specific destination.
Limits are a lot like that. They're not always about where you actually land, but about the destination you were approaching. In this lesson, we'll learn how to look at a function's graph—its road map—and determine that intended destination.
Concept overview
flowchart TD
A[Find lim f(x) as x->c] --> B{Check left-hand limit L-}
A --> C{Check right-hand limit L+}
B --> D{Is L- a finite number?}
C --> E{Is L+ a finite number?}
D -- No --> F[Limit DNE: Unbounded or Oscillating]
E -- No --> F
D -- Yes --> G{Are L- and L+ equal?}
E -- Yes --> G
G -- No --> H[Limit DNE: Jump]
G -- Yes --> I[Limit Exists and equals L- = L+]
Core explanation
Hello there. I'm Saavi, and I'm glad you're here. Let's talk about one of the most foundational ideas in all of calculus: the limit. For now, we're going to forget about complicated equations and just use our eyes.
What is a Limit, Visually?
A limit is the y-value a function approaches as the x-value gets closer and closer to some number. Think of the graph of a function, f(x), as a path. We want to know the elevation (the y-value) we're approaching as we walk toward a specific landmark (the x-value).
We write this in math notation as:
lim_{x→c} f(x) = L
This is read as "the limit of f(x) as x approaches c is L."
cis the x-value you're heading towards.Lis the y-value your function is getting infinitely close to.
Imagine two friends, Priya and Marcus, who agree to meet on a hiking trail at the "2-mile marker" (our x = 2). Priya is hiking from the 1-mile marker, and Marcus is hiking from the 3-mile marker. The limit is the elevation they are both approaching as they get to the 2-mile marker.
The Two-Sided Approach: Left-Hand and Right-Hand Limits
For the limit to exist, Priya and Marcus must be heading towards the exact same spot. This gives us the single most important idea for finding limits from a graph: you must check the approach from both sides.
Left-Hand Limit: This is the value the function approaches as x gets close to c from the left side (from values smaller than c). This is Priya's path.
We write it with a little minus sign: lim_{x→c⁻} f(x)
Right-Hand Limit: This is the value the function approaches as x gets close to c from the right side (from values larger than c). This is Marcus's path.
We write it with a little plus sign: lim_{x→c⁺} f(x)
For the overall limit L to exist, the left-hand limit must equal the right-hand limit.
lim_{x→c⁻} f(x) = lim_{x→c⁺} f(x) = L
If they are heading towards different elevations, they won't meet up, and we say the limit does not exist.
The Limit is NOT the Function's Value
Look at this graph.
Imagine a graph with a smooth curve. At x=3, there is an open circle (a "hole") at y=4. There is also a single, separate solid dot at (3, 1).- As you trace the graph from the left towards
x=3, your finger gets closer and closer toy=4. So,lim_{x→3⁻} f(x) = 4. - As you trace from the right towards
x=3, your finger also gets closer and closer toy=4. So,lim_{x→3⁺} f(x) = 4. - Since both sides agree, the overall limit is 4:
lim_{x→3} f(x) = 4.
But what is the actual value of the function at x=3? The solid dot tells us that f(3) = 1.
The limit is 4, but the function's value is 1. They are different, and that is perfectly okay! Remember the GPS analogy: the intended destination was the house (y=4), even if a roadblock forced you to park somewhere else (y=1).
When Limits Don't Exist (DNE)
Sometimes, a limit simply does not exist (we often write DNE). The AP exam expects you to know the three main reasons why.
1. The Jump: The left-hand limit does not equal the right-hand limit.
Imagine our hikers, Priya and Marcus, approach the 2-mile marker, but Priya is heading for a spot at an elevation of 100 feet and Marcus is heading for a spot at 150 feet. They aren't meeting. On a graph, this looks like a "jump." Because lim_{x→c⁻} f(x) ≠ lim_{x→c⁺} f(x), the overall limit DNE.
2. Unbounded Behavior: The function flies up to positive infinity or dives down to negative infinity.
This happens at a vertical asymptote. As your x-value gets closer to c, the y-value gets huge without any bound. Since the function isn't approaching one specific, finite number, the limit does not exist. (On the AP Exam, you might be asked to be more specific and say the limit is ∞ or -∞, but the formal limit DNE).
3. Oscillating Behavior: The function fluctuates between two values faster and faster as it approaches c.
The classic example is f(x) = sin(1/x) as x approaches 0. The graph goes wild, bouncing up and down between -1 and 1. It never settles on a single y-value. The limit DNE.
A Word of Caution: Graphs Can Be Deceiving
A graph is a wonderful tool, but it has limits (pun intended!). Because of issues of scale, a graph might hide important details. A function could look perfectly smooth, but if you were to zoom in a million times, you might find a tiny hole or a rapid oscillation.
For the AP exam, the graphs you're given will be clear and designed to be read accurately. But it's a good habit to remember that a picture doesn't always tell the whole story. That's why we'll learn algebraic techniques to confirm what we see.
Worked examples
Let's put this all into practice with a typical AP-style problem.
The Piecewise Function
Consider the graph of the function h(x) below.
Imagine a graph of h(x).
- From the left, a line segment goes to an open circle at (-2, 3).
- A solid dot exists at (-2, 1).
- From x=-2 to x=4, a parabola opens upwards, starting from the open circle at (-2, 3) and going through (0, -1) to a solid dot at (4, 3).
- To the right of x=4, a horizontal line starts at y=3.Problem: Find the following:
a) lim_{x→-2} h(x)
b) h(-2)
c) lim_{x→4} h(x)
d) h(4)
Solution:
Part (a): lim_{x→-2} h(x)
- WhyTo find the overall limit, we must check the approach from both the left and the right.
- Step 1: Left-hand limitTrace the graph from the left side towards
x=-2. Your finger will follow the line segment up to the open circle. The y-value you are approaching is 3. So,lim_{x→-2⁻} h(x) = 3. - Step 2: Right-hand limitTrace the graph from the right side (starting somewhere like
x=0) back towardsx=-2. Your finger will follow the parabola down to the open circle. The y-value you are approaching is also 3. So,lim_{x→-2⁺} h(x) = 3. - Step 3: ConclusionSince the left-hand limit (3) equals the right-hand limit (3), the overall limit exists and is 3.
- Answer
lim_{x→-2} h(x) = 3.
Part (b): h(-2)
- WhyThis is asking for the actual value of the function at
x=-2, not the limit. We need to find the solid dot at that x-value. - Step 1Look at the vertical line
x=-2. There is an open circle aty=3and a solid dot aty=1. The solid dot defines the function's value. - Answer
h(-2) = 1. - Common Mistake AlertMany students will say the limit is 1, or that
h(-2)is 3. Notice how the limit (the approach) and the value (the actual point) are different here!
Part (c): lim_{x→4} h(x)
- WhySame process. Check both sides.
- Step 1: Left-hand limitTrace the parabola towards
x=4. Your finger approaches the solid dot at(4, 3). The y-value is 3. So,lim_{x→4⁻} h(x) = 3. - Step 2: Right-hand limitTrace the horizontal line from the right back towards
x=4. Your finger stays at a constant y-value of 3. So,lim_{x→4⁺} h(x) = 3. - Step 3: ConclusionBoth sides agree.
- Answer
lim_{x→4} h(x) = 3.
Part (d): h(4)
- WhyFind the solid dot at
x=4. - Step 1At
x=4, we have a solid dot aty=3. - Answer
h(4) = 3. In this case, the limit and the function's value happen to be the same.
Try it yourself
Let's test your skills. Look at the graph of g(x) below and find the requested values.
Imagine a graph of g(x).
- There is a vertical asymptote at x = 1.
- To the left of x=1, the graph goes up towards +∞ as x approaches 1.
- To the right of x=1, the graph comes down from +∞.
- There is a hole at (5, 2).
- The function is defined at f(5) = 6 (a solid dot).Problem:
- Find
lim_{x→1⁺} g(x) - Find
lim_{x→1} g(x) - Find
lim_{x→5} g(x) - Find
g(5)
Hints:
- For #1, what y-value is the function approaching as you trace it from the right side towards
x=1? - For #2, what does it mean if the function goes to infinity? Does the limit exist in the formal sense?
- For #3 and #4, remember the crucial difference between the limit (the approach) and the function's value (the solid dot).
In simple terms, this topic is about looking at a function's graph to predict what value it's getting closer and closer to, even if it never actually gets there.
Imagine a graph with a smooth curve. At x=3, there is an open circle (a "hole") at y=4. There is also a single, separate solid dot at (3, 1).- LIM-1.C: Estimate limits of functions.
- LIM-1.C.1
- The concept of a limit includes one sided limits.
- LIM-1.C.2
- Graphical information about a function can be used to estimate limits.
- LIM-1.C.3
- Because of issues of scale, graphical representations of functions may miss important function behavior.
- LIM-1.C.4
- A limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.
flowchart TD
A[Find lim f(x) as x->c] --> B{Check left-hand limit L-}
A --> C{Check right-hand limit L+}
B --> D{Is L- a finite number?}
C --> E{Is L+ a finite number?}
D -- No --> F[Limit DNE: Unbounded or Oscillating]
E -- No --> F
D -- Yes --> G{Are L- and L+ equal?}
E -- Yes --> G
G -- No --> H[Limit DNE: Jump]
G -- Yes --> I[Limit Exists and equals L- = L+]
Read what Saavi narrates
Hi everyone, I'm Saavi, and welcome to Shrutam. Let's talk about one of my favorite topics, limits.
Think about this: you're driving from Dallas to a friend's house in Austin. Your GPS says you'll arrive at 3:00 PM. But right when you get there, you see a roadblock on their street and have to park a block away. You didn't technically arrive *at* the house in your car... but your entire trip was aimed at that one specific destination, right?
That's a limit. It's the intended destination, not always where you actually land.
In calculus, we're learning to follow a function's path on a graph from both the left and the right. This helps us predict the y-value the function is heading towards at any given x-value.
Let's look at an example. Imagine a graph of a function, let's call it h of x. At x equals negative two, there's a hole in the graph at y equals three. But there's also a separate, solid dot at y equals one.
So, what's the limit as x approaches negative two?
Well, let's check both sides. If you trace the graph from the left, your finger gets closer and closer to a y-value of three. If you trace from the right, your finger *also* gets closer to that same y-value of three. Since both sides agree, the limit is three.
But, what is the function's value, h of negative two? For that, we look for the solid dot. The solid dot is at y equals one. So h of negative two is one.
This brings us to the most common mistake students make: confusing the limit with the function's value. The limit was three, but the actual value was one. They can be different! The limit is the approach... the value is the actual landing spot. Always remember to check the path from both sides for the limit, and look for the solid dot for the function's value.
You're building a fantastic foundation here. Keep practicing, and you'll be reading graphs like a pro in no time.
The limit is the value the function *approaches*, while `f(2)` is the value the function *is* at `x=2`. They can be different (like at a hole with a dot elsewhere).
Always find the limit by checking the approach from the left and right. Find the function's value by looking for the solid dot at that exact x-value.
A hole just means `f(c)` is undefined or defined elsewhere. As long as the path from the left and the path from the right are heading to the same y-value (the location of the hole), the limit exists.
Remember the limit cares about the journey, not the destination's existence. If both sides approach the same y-value of the hole, that y-value is your limit.
This confuses limits with differentiability (a topic for later!). As `x` approaches 0 from the left, `y` approaches 0. As `x` approaches 0 from the right, `y` also approaches 0. Since the left and right limits match, the limit is 0.
Apply the two-sided test. If the left-hand limit equals the right-hand limit, the limit exists, regardless of how "pointy" the graph is.
The overall limit only exists if BOTH one-sided limits exist and are equal. Ignoring one side can lead you to the wrong conclusion at a jump discontinuity.
Make it a habit. To find `lim_{x→c}`, always check `lim_{x→c⁻}` and `lim_{x→c⁺}` first.
While technically the limit does not exist because it's not a finite number, the AP exam often wants you to describe the behavior.
If the function goes up on both sides of the asymptote, the best answer is `∞`. If it goes down, `-∞`. If it goes up on one side and down on the other, then "DNE" is the most appropriate answer.