Estimating a finite limit from a table of values

As $x$ approaches $0$ from the left (negative values getting closer to $0$) and from the right (positive values getting closer to $0$), the table shows $g(x)$ getting closer and closer to $1$, with the values $1.00335, 1.0000333, 1.000000333, \ldots$ shrinking toward $1$ from both directions.

The graph confirms this: the curve dips down and appears to pass through the height $y=1$ right around $x=0$, even though there's no actual point plotted at $x=0$ since $g$ is undefined there. Since both the left-hand and right-hand trends agree on the same value, we conclude the two-sided limit exists and equals that common value.

Therefore $\displaystyle\lim_{x\to 0}\frac{\tan(x)}{x} = 1$.

No. The function $g(x) = \dfrac{\tan(x)}{x}$ is undefined at $x = 0$, since that would require dividing by zero. The table and graph only show values for $x$ near zero, never at zero itself. Despite this, the limit as $x$ approaches zero still exists and equals one, since limits depend only on nearby behavior, not on the value at the point itself.