Estimating Limits from Graphs

The Visual Approach

To find $\lim_{x \to a} f(x)$ from a graph:

1. Locate x = a on the horizontal axis
2. Trace from the left - Where is the curve heading as you approach a from the left?
3. Trace from the right - Where is the curve heading as you approach a from the right?
4. Check if they agree - Do both sides point to the same y-value?

Key Visual Clues

Case 1: Continuous Point

If the function is smooth and connected at x = a, the limit exists and equals f(a).

The curve flows smoothly through the point - both sides meet at the same spot.

Case 2: Hole (Removable Discontinuity)

Even if there's a hole at x = a, the limit can still exist!

Follow where the curve would go if the hole weren't there. Both sides still point to the same y-value.

Case 3: Jump Discontinuity

If the left side approaches one value and the right side approaches a different value, the limit does not exist.

The curve "jumps" - there's no single value both sides agree on.

Case 4: Vertical Asymptote

If the function shoots up to $\infty$ or down to $-\infty$, we say the limit is infinite (or DNE, depending on context).

The curve races off toward infinity - unbounded behavior.

Reading One-Sided Limits

Left-hand limit $\lim_{x \to a^-} f(x)$:

• Trace along the curve from the left
• What y-value does it approach?

Right-hand limit $\lim_{x \to a^+} f(x)$:

• Trace along the curve from the right
• What y-value does it approach?

Remember: If left ≠ right, the two-sided limit DNE!

Important Graph Features

Practice Strategy

When looking at a graph:

1. Cover the point in question with your finger
2. Slide toward it from the left - where are you heading?
3. Slide toward it from the right - where are you heading?
4. If both sides agree, that's your limit!

Example Interpretation

Imagine a graph where:

• As you approach x = 3 from the left, the y-values approach 5
• As you approach x = 3 from the right, the y-values approach 5
• But there's an open circle at (3, 5) and a solid dot at (3, 7)

What's the limit?

$\lim_{x \to 3} f(x) = 5$ (even though f(3) = 7!)

The limit is about where the function is heading, not where it is.

Since both one-sided limits exist and are equal:

$\lim_{x \to 2} f(x) = 4$

Comparing to f(2):

We're told that f(2) = 1.

Therefore: $\lim_{x \to 2} f(x) = 4 \neq f(2) = 1$

The limit does NOT equal the function value!

What this looks like:

• There's a hole at (2, 4) where the function "wants" to be
• There's a solid dot at (2, 1) where the function actually is
• This is called a removable discontinuity