Module 1.1 Content

The Numerical Approximation

How to estimate the value of a limit by constructing a table of values — approaching from both sides without ever landing on the point itself.

1

The Idea & Step-by-Step Method

When we want to evaluate $\lim_{x \to a} f(x)$, we are asking: "What value does $f(x)$ approach as $x$ gets arbitrarily close to $a$?"

The numerical approach answers this by plugging in values of $x$ that are incredibly close to $a$ — from both the left and the right — and observing the pattern of the outputs.

📝 Step-by-Step Method
  1. 1

    Choose values approaching from the left

    Pick 3–4 values slightly less than $a$ (e.g. if $a=3$: $2.9,\;2.99,\;2.999$).

  2. 2

    Choose values approaching from the right

    Pick 3–4 values slightly greater than $a$ (e.g. $3.001,\;3.01,\;3.1$).

  3. 3

    Evaluate $f(x)$ at each value

    Plug each $x$ into $f(x)$ and record the results in a table.

  4. 4

    Identify the trend

    If both sides converge to the same number $L$, then $\lim_{x \to a} f(x) = L$.

2

Example 1 — Rational Function

Evaluate the following limit numerically:

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$$

Plugging $x = 3$ directly gives $\tfrac{0}{0}$ — undefined. Instead, let's approach $3$ from both sides.

$x$ $2.9$ $2.99$ $2.999$ $\to 3 \leftarrow$ $3.001$ $3.01$ $3.1$
$f(x)$ $5.9$ $5.99$ $5.999$ Undef. $6.001$ $6.01$ $6.1$
$x \to 3^-$  (from left) $x \to 3^+$  (from right)

Both sides converge to $6$. Therefore:

$$\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = 6$$
💡 Key Insight

Even though $f(3)$ is undefined, the limit is perfectly well-defined and equals $6$. The limit only cares about what happens near $x = 3$, not at it.

3

Example 2 — A Famous Trigonometric Limit

Let's evaluate one of the most important limits in all of calculus. Make sure your calculator is set to Radians.

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

Evaluating at $x=0$ gives $\tfrac{0}{0}$. Let's build the table:

$x$ $-0.1$ $-0.01$ $-0.001$ $\to 0 \leftarrow$ $0.001$ $0.01$ $0.1$
$f(x)$ $0.99833$ $0.99998$ $0.99999$ Undef. $0.99999$ $0.99998$ $0.99833$
$x \to 0^-$  (from left) $x \to 0^+$  (from right)

From both sides, $f(x)$ is clearly converging to $1$. This result is so fundamental it is used throughout trigonometry, physics, and engineering:

$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$
⚠️ Limitation of This Method

The numerical approach gives us a strong estimate, but it is not a rigorous proof. Tables can be misleading for functions that behave erratically. Always verify with an analytical method when precision matters.

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