Topic 1.3 Notes – Estimating Limit Values from Graphs

1. What a Limit Is From a Graph

When you see

$$\lim_{x \to a} f(x) = L$$

it means:

As x gets close to a, the y-values get close to L.

It does not mean $f(a) = L$.

From a graph, you ignore the exact value at $x=a$ and look at what the graph is doing nearby.

One-Sided Limits

Sometimes the graph behaves differently from each side.

• Left-hand limit: $\lim_{x \to a^-} f(x)$
  Approach from values less than a (move rightward toward a).
• Right-hand limit: $\lim_{x \to a^+} f(x)$
  Approach from values greater than a (move leftward toward a).

The two-sided limit exists only if:

$$\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$$

If those don’t match, the limit is DNE.

Important: the limit can exist even if

• There’s a hole at $x=a$
• The function isn’t defined there
• The filled dot is at a totally different height

The graph’s approach is what matters.

2. Estimating a Limit From a Graph

When this shows up on a quiz or no-calculator MCQ, they want you to read behavior carefully.

Here’s the process in your head:

1. Find the x-value you’re approaching.
2. Trace from the left. What y-value are you getting close to?
3. Trace from the right. What y-value are you getting close to?
4. Compare.
   • Same number → that’s the limit.
   • Different numbers → DNE.
   • Growing without bound → infinite behavior (also DNE as a finite number).

Common Graph Situations

Here are three patterns you’ll see over and over:

Hole, jump discontinuity, and vertical asymptote

Interpret them like this:

• Hole (open circle) → limit usually exists because both sides approach the same y-value.
• Jump → left and right limits differ → DNE.
• Vertical asymptote → function is unbounded near that x-value.

And if there’s a filled dot somewhere else? Ignore it for the limit.

One mistake I see constantly: students read the filled dot instead of the approaching value. That costs easy points.

3. When a Limit Does Not Exist

There are exactly three patterns you need to recognize.

1. Left and Right Don’t Match (Jump)

Example shape:

• From the left, graph approaches 4.
• From the right, graph approaches −1.

Since 4 ≠ −1, the limit DNE.

A classic algebra example is

$$\lim_{x \to 0} \frac{|x|}{x}$$

Left side gives −1, right side gives 1. No agreement.

2. Unbounded Behavior (Infinite)

If the function shoots upward or downward near a value, it’s unbounded.

Examples:

$$\lim_{x \to 0} \frac{1}{x} \quad \text{DNE}$$

$$\lim_{x \to 0} \frac{1}{x^2} = \infty$$

Even if both sides go to $+\infty$, the limit does not exist as a finite number.

On FRQs, writing “approaches infinity” is fine to describe behavior, but technically the limit DNE.

3. Oscillation

Some functions never settle.

Example:

$$\lim_{x \to 0} \sin\left(\frac{1}{x}\right)$$

Near 0, the graph keeps bouncing between −1 and 1 faster and faster.

It never locks in on a single value as $x$ approaches 0.

Since it never approaches one number, the limit DNE.

4. Graph Scale and Hidden Behavior

Graphs can lie to you.

Scale Issues

If the window is too wide:

• A vertical asymptote might look like a point.
• Oscillation might look like a blur.
• A tiny jump might look continuous.

AP exam graphs are usually trustworthy, but you still read them carefully.

If algebra suggests something weird could happen, don’t blindly trust a smooth-looking picture.

Especially on calculator sections, window settings can hide behavior. Zooming mentally near the x-value matters.

5. Common Mistakes

• Confusing $f(a)$ with $\lim_{x \to a} f(x)$
• Checking only one side
• Calling infinity a number
• Looking at the point instead of the approach
• Deciding from far away instead of zooming in near $a$

When you answer, always think:

Left?
Right?
Same?
Finite?

That logic is what graders look for on written responses.