Calculus I · Limits and Continuity · lesson

How to Read Limits From a Graph

A Reliable Method for Reading a Graph

Graph questions are often rushed and surprisingly easy to misread. At each marked input $x=a$ , use the same order.

Method

Graph-Reading Routine

• Trace the curve toward $a$ from the left. Record the height approached. • Trace the curve toward $a$ from the right. Record the height approached. • Compare the one-sided limits. • Only then inspect the filled point to find $f(a)$ .

Worked example

Reading four quantities from one graph

Suppose a graph has an open circle at $(3,6)$ , the curve approaches that circle from both sides, and a filled point appears at $(3,-1)$ . Find

\lim_{x\to3^-}f(x),\quad \lim_{x\to3^+}f(x),\quad \lim_{x\to3}f(x),\quad f(3).

Show worked solution

Following the curve from the left gives

\lim_{x\to3^-}f(x)=6.

Following the curve from the right gives

\lim_{x\to3^+}f(x)=6.

The two sides agree, so

\lim_{x\to3}f(x)=6.

The filled point gives the actual value:

f(3)=-1.

Thus,

\boxed{6,\ 6,\ 6,\ -1}.

After the explanation

Use the section idea

Reading lens

What are nearby outputs doing as the input approaches the target from both sides?

Mental model

Imagine tightening a window around the target input and watching where all nearby outputs are forced to gather.

Decision

Read the left-hand and right-hand behavior separately first; combine them only after both sides approach the same output.

Common trap

The function value at the target can be missing or deliberately moved, so never substitute a plotted dot for evidence from both sides.

Check yourself

You understand the section when you can explain a limit from a graph, table, and sentence without confusing it with the function value.

Source & rights

Original instruction with traceable references.

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