Topic 1.4 Notes – Estimating Limit Values from Tables

What a Limit Is

When you see

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

it means:

As x gets closer and closer to $a$, the function values get closer and closer to $L$.

Three things to keep straight:

• The function does not need to be defined at $x = a$.
• $f(a)$ might equal the limit, might be different, or might not exist.
• A limit is about near, not at.

If you plug in $a$ directly and get a normal real number, that’s usually the limit (assuming the function is continuous there). If you plug in and get something undefined like $0/0$, that doesn’t mean the limit doesn’t exist. It means you need another approach. That’s where tables come in.

One-Sided Limits

Sometimes behavior from each side matters.

• $\lim_{x \to a^-} f(x)$ means approach from the left (values less than $a$).
• $\lim_{x \to a^+} f(x)$ means approach from the right (values greater than $a$).

The full limit exists only if both sides match:

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

If the two sides approach different numbers, the limit does not exist (DNE).

Here’s what that looks like visually at $x = 2$:

Jump discontinuity at $x = 2$

Even if both sides are perfectly well-behaved, if they approach different heights, there is no two-sided limit.

Estimating a Limit from a Table

When you’re given a table, you’re using numerical evidence to decide what value the function is approaching.

Step-by-step process

1. Check values on both sides of $a$
   Look at x-values slightly less than and slightly greater than the target.

   Example pattern:

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

2. Look at the corresponding $f(x)$ values
   Ignore $f(a)$ if it’s listed. That’s not what we’re studying.

3. Watch the trend
   Ask:

   • Are both sides settling toward the same number?
   • Are they separating?
   • Are they getting very large positive or negative?

The limit is the number both sides are squeezing toward.

Here’s what a “good” table looks like when a limit exists as $x \to 5$:

Table of values approaching 5 from both sides

Both sides tighten around 8. So

$$\lim_{x \to 5} f(x) = 8.$$

Possible Table Behaviors

You only need to recognize a few patterns.

1. Both sides approach the same finite number

✔️ The limit exists and equals that number.

Even if the function never actually equals it.

2. Left and right approach different numbers

❌ The limit does not exist. This often signals a jump discontinuity.

3. Values grow without bound

If numbers explode upward,

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

If they drop without bound,

$$\lim_{x \to a} f(x) = -\infty.$$

That’s vertical asymptote behavior. On AP problems, you may state the limit is $\infty$ or $-\infty$.

4. Oscillating or unstable values

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

When Tables Are Used

Tables show up when:

• You only have numerical data.
• Direct substitution gives $0/0$.
• A function is undefined at the point.
• A problem asks you to justify a limit using numerical evidence.

On multiple choice, you’ll scan quickly for trends. On free response, you may need to say something like:
“As x approaches 3 from both sides, the values of $f(x)$ approach 4. Therefore, the limit is 4.”

That justification matters. You’re connecting data to the definition of a limit.