Topic 1.4 Notes – Estimating Limit Values from Tables

1. What It Means to Estimate a Limit from a Table

A limit describes the value a function approaches as $x$ approaches some number $a$.

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

This means: as $x$ gets close to $a$, the outputs get close to $L$.

Two reminders that matter a lot on tests:

• The limit depends on values near $a$, not necessarily at $a$.
• $f(a)$ might be different from the limit - or undefined - and the limit can still exist.

When you’re estimating from a table:

• You look at $x$-values very close to $a$.
• You check what the corresponding $f(x)$-values are doing.
• You decide what number those outputs are trending toward.

You’re not looking for equality. You’re looking for behavior.

This often shows up when direct substitution would give something like $0/0$, or when you’re only given numerical data instead of a formula.

2. One-Sided Limits

When estimating from a table, you almost always need to think about both sides of $a$.

Left-hand limit

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

• $x$ values less than $a$
• Moving toward $a$ from the left

Right-hand limit

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

• $x$ values greater than $a$
• Moving toward $a$ from the right

Here’s the rule you need to know cold:

$$\lim_{x \to a} f(x) \text{ exists only if } \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$$

If the two sides:

• ✔️ Approach the same number → the limit exists.
• ❌ Approach different numbers → the limit does not exist.
• ❌ Grow without bound ($+\infty$ or $-\infty$) → infinite limit (vertical asymptote behavior).

On multiple-choice questions, they love hiding a mismatch between left and right in small decimal differences. Always check both sides.

3. How to Estimate from a Table

Let’s say you’re given a table near $x = 2$.

Table of values approaching $x = 2$ from both sides

This is exactly the kind of setup you’ll see on multiple choice and FRQs.

Here’s how to read it.

1. Identify the target

The limit is as $x \to 2$.

2. Separate left and right

• Left side: 1.9, 1.99, 1.999
• Right side: 2.001, 2.01, 2.1

3. Look for a pattern

From the left, outputs: 4.1 → 4.01 → 4.001
From the right, outputs: 3.9 → 3.99 → 3.999

Both sides are getting closer to 4.

So,

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

Notice something important: none of the outputs actually equal 4. That does not matter. The limit is about approach.

On FRQs, if they say “estimate,” give a reasonable decimal and justify it by referencing both sides.

4. Recognizing Different Behaviors

Tables can show different limit situations. You need to recognize these patterns quickly.

Removable Discontinuity (Hole)

• Table values approach a finite number.
• The function might be undefined at that point.
• Limit exists.

This is common when substitution gives $0/0$.

Jump Discontinuity

• Left side approaches one number.
• Right side approaches a different number.
• Limit does not exist.

If a table showed left approaching 3 and right approaching 5, that’s immediately DNE.

Vertical Asymptote

• Values grow very large positive or negative.
• Outputs trend toward $\pm \infty$.

That means an infinite limit.

No Clear Pattern

If values bounce around and don’t settle toward anything, the limit does not exist.