Reading limits from graphs and tables | Code Scholars Tutoring

Understand the idea

A limit describes the output approached as the input approaches a location; the value at that location can be different or undefined. A two-sided finite limit exists only when both one-sided limits agree. Tables support a conjecture but do not prove behavior between entries. On a graph follow the branches, rather than the filled point. Interpret average rates over shrinking intervals as motivation for instantaneous change.

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

Worked example

A graph approaches $\displaystyle 4$ from both sides at $\displaystyle x=2$ , but has a filled point at $\displaystyle (2,7)$ . Find the limit and function value.

Both branches approach the same height, so

\displaystyle \lim_{x\to2}f(x)=4

. The filled point gives

\displaystyle f(2)=7

. The mismatch makes the function discontinuous, even though the limit exists.

Try it yourself

Left-hand values approach $\displaystyle -1$ and right-hand values approach $\displaystyle 3$ at $\displaystyle x=0$ . What is the two-sided limit?

Show solution and reasoning

It does not exist: the two one-sided limits are unequal. This is a jump, not an infinite limit.