A Limit at a Hole

Use a table to estimate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$.

Check the point itself: At $x = 2$: $\frac{4 - 4}{2 - 2} = \frac{0}{0}$ = $f(2)$ is undefined, so we need a table

Approach from the left: $f(1.9) = 3.9$, $f(1.99) = 3.99$, $f(1.999) = 3.999$ = Left side heads toward $4$

Approach from the right: $f(2.1) = 4.1$, $f(2.01) = 4.01$, $f(2.001) = 4.001$ = Right side heads toward $4$

Compare both sides: Both sides approach the same value = $\lim_{x \to 2} f(x) = 4$

Answer: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$, even though $f(2)$ does not exist

A Limit You Cannot Factor

Estimate $\lim_{x \to 0} \frac{\sin x}{x}$ ($x$ in radians).

Set the calculator to radians: Limits with $\sin x$ always use radians = Mode: RAD

Approach from the right: $x = 0.1$: $0.998334$; $x = 0.01$: $0.999983$; $x = 0.001$: $0.9999998$ = Heading toward $1$

Approach from the left: $\frac{\sin(-x)}{-x} = \frac{-\sin x}{-x} = \frac{\sin x}{x}$, so $x = -0.1$ gives $0.998334$ again = Same values, heading toward $1$

Conclude: Both sides approach $1$ = $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Answer: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

When the Two Sides Disagree

Use a table to decide whether $\lim_{x \to 0} \frac{|x|}{x}$ exists.

Approach from the left: $x = -0.1$: $\frac{0.1}{-0.1} = -1$; $x = -0.01$: $-1$; $x = -0.001$: $-1$ = Left side stays at $-1$

Approach from the right: $x = 0.1$: $\frac{0.1}{0.1} = 1$; $x = 0.01$: $1$; $x = 0.001$: $1$ = Right side stays at $1$

Compare both sides: $-1 \neq 1$ = The sides approach different numbers

Conclude: A two-sided limit needs one single value = $\lim_{x \to 0} \frac{|x|}{x}$ does not exist

Answer: The limit does not exist: the left side approaches $-1$ and the right side approaches $1$

Outputs That Grow Without Bound

Use a table to describe $\lim_{x \to 3} \frac{1}{(x - 3)^2}$.

Approach from the left: $x = 2.9$: $\frac{1}{0.01} = 100$; $x = 2.99$: $10\,000$; $x = 2.999$: $1\,000\,000$ = Values explode upward

Approach from the right: $x = 3.1$: $100$; $x = 3.01$: $10\,000$; $x = 3.001$: $1\,000\,000$ = Same explosion upward

Look for a pattern: Each time $x$ gets $10$ times closer to $3$, $f(x)$ gets $100$ times bigger = No number is being approached

Describe the behavior: The outputs grow without bound from both sides = $\lim_{x \to 3} \frac{1}{(x - 3)^2} = \infty$

Answer: The function grows without bound: $\lim_{x \to 3} \frac{1}{(x - 3)^2} = \infty$, so there is no finite limit (vertical asymptote $x = 3$)

When a Table Fools You

A student uses $x = 0.1$, $0.01$, $0.001$ to estimate $\lim_{x \to 0} \sin\left(\frac{\pi}{x}\right)$ and gets $0$ every time. Is the limit $0$?

Check the student's values: $\sin(10\pi) = 0$, $\sin(100\pi) = 0$, $\sin(1000\pi) = 0$ = All zeros, which looks convincing

Try other values close to $0$: $x = 0.4$: $\sin(2.5\pi) = 1$; $x = \frac{2}{9} \approx 0.222$: $\sin(4.5\pi) = 1$ = Values equal to $1$ appear

Try even more values: $x = \frac{2}{3}$: $\sin(1.5\pi) = -1$; $x = \frac{2}{7}$: $\sin(3.5\pi) = -1$ = Values equal to $-1$ appear too

Conclude: Near $0$ the function keeps swinging between $-1$ and $1$ = The limit does not exist

Answer: No. $\sin\left(\frac{\pi}{x}\right)$ oscillates between $-1$ and $1$ infinitely often near $0$, so the limit does not exist. The chosen $x$ values just happened to hit zeros.

Mistake: Using only one side of $a$

Why: A table with only $x = 1.9$, $1.99$, $1.999$ shows the left-hand limit. The right side may head somewhere else, as with $\frac{|x|}{x}$.

Correct: Always build both columns: values just below $a$ and values just above $a$.

Mistake: Choosing $x$ values that are not close to $a$

Why: Values like $x = 1$, $1.5$, $2.5$, $3$ are too far away. The function can behave very differently far from the point.

Correct: Move closer by a factor of $10$ each time: $a \pm 0.1$, $a \pm 0.01$, $a \pm 0.001$.

Mistake: Putting $x = a$ in the table and using $f(a)$ as the limit

Why: The limit describes what happens near $a$, not at $a$. At a hole $f(a)$ may be undefined, and at a moved point it may be a different number.

Correct: Leave $x = a$ out of the table and read only the trend of the nearby values.

Mistake: Trusting a few values blindly

Why: Special choices of $x$ can hide oscillation, as with $\sin\left(\frac{\pi}{x}\right)$, and rounding can hide small differences between the sides.

Correct: Use several values that are not all of the same pattern, keep enough decimals, and confirm with a graph or algebra.

Instantaneous Speed in Physics

A speedometer shows speed at one instant. Physicists find it by computing average speeds over shorter and shorter time intervals and reading the trend in the table.

A ball falls $s(t) = 5t^2$ meters in $t$ seconds. Average speeds starting at $t = 2$: over $0.1$ s it is $20.5$ m/s, over $0.01$ s it is $20.05$ m/s, over $0.001$ s it is $20.005$ m/s.

Growth Rates in Biology

A bacteria culture that doubles every hour does not grow in jumps. Its instantaneous growth rate is found from a table of the expression $\frac{2^h - 1}{h}$ for tiny time steps $h$.

For $h = 0.1$ hours the table gives $0.7177$, for $h = 0.01$ it gives $0.6956$, for $h = 0.001$ it gives $0.6934$: the rate settles near $0.693$ per hour.

Signal Processing and Audio

Audio software uses functions like $\frac{\sin(2x)}{x}$ to filter sound. At $x = 0$ the formula gives $\frac{0}{0}$, so programmers use the limit from a table to fill in the missing value.

Near $0$ (radians), $\frac{\sin(2x)}{x}$ gives $1.98669$ at $x = 0.1$ and $1.99987$ at $x = 0.01$.

Q: How many values do I need in the table?

A: Usually three on each side, such as $a \pm 0.1$, $a \pm 0.01$, $a \pm 0.001$, are enough to see a clear trend. If the trend is not clear, add values that are even closer, like $a \pm 0.0001$.

Q: Why should I not just plug in $x = a$?

A: Because the limit is about the values near $a$, not at $a$. Often $f(a)$ is undefined (a hole or $\frac{0}{0}$), and sometimes it exists but differs from the limit. The table shows the trend that $f(a)$ cannot show.

Q: Is a table a proof that the limit is correct?

A: No. A table only gives strong evidence. Exact answers come from algebra (Limits by Direct Substitution, factoring) or from limit laws. Tables are excellent for estimating and for checking an algebraic answer.

Q: What does it mean when my calculator shows strange values very close to $a$?

A: With extremely small differences like $x - a = 0.0000000001$, a calculator can lose digits when it subtracts nearly equal numbers. If the values suddenly jump around, step back to $a \pm 0.001$ or $a \pm 0.0001$, where the arithmetic is still reliable.