6.1 · Limits from graphs, tables, and algebra

A limit describes approach, not evaluation

The statement $\displaystyle \lim_{x\to a}f(x)=L$ concerns outputs near $\displaystyle a$, with the input distinct from $\displaystyle a$. A missing or different point value does not automatically prevent the limit.

Trace the graph toward the input from both sides. A filled dot tells you the defined point value; the nearby curve tells you the approached height.

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

One-sided limits must agree

A finite two-sided limit exists exactly when the left and right limits exist and agree. If the branches approach different heights, the two-sided limit does not exist regardless of the point value.

A finite table suggests a trend but does not prove behavior at every nearby input. Combine numerical evidence with a formula or a known graph whenever possible.

Zero over zero asks for more analysis

For $\displaystyle (x^2-4)/(x-2)$ near $\displaystyle 2$, substitution gives the indeterminate form $\displaystyle 0/0$. Factoring and canceling for $\displaystyle x\ne2$ yields $\displaystyle x+2$, whose limit is $\displaystyle 4$.

The cancellation does not define the original expression at the excluded point. A limit can exist even when the original function is undefined there.