Connecting Multiple Representations of Limits

One limit, four disguises

• A single limit can appear as a graph, a table, a formula, or a sentence in words.
• $\displaystyle\lim_{x\to2}f(x)=3$ is the same fact whether you see a curve heading to $3$, a table closing in on $3$, an algebra result of $3$, or "$f$ approaches $3$ near $2$."
• Fluency means moving between these four representations 表示 without losing the meaning.
• The exam mixes them on purpose — so practice the translation both ways.

Graph ⇄ words

• From a graph 图像: trace both sides toward $c$ and read the shared height.
• Into words: "the left and right limits agree at $3$, so the limit is $3$" — even if the plotted point sits elsewhere.
• A jump on the graph becomes "the one-sided limits differ, so the limit does not exist."
• Practice describing a graph's behavior in one precise sentence.

One-sided limits disagree: from the left f approaches 1 and from the right 3 so the limit at 0 does not exist

Table ⇄ formula

• From a table 表格: watch the outputs converge; that shared number is your numerical estimate.
• Confirm it with the formula by factoring or substituting — algebra turns the estimate into certainty.
• If the table and the algebra agree, you have strong, cross-checked evidence.
• If they disagree, recheck your inputs — a table can be fooled, algebra usually cannot.

Let representations check each other

• The real skill: use one representation to confirm a limit found from another.
• Found $L$ by algebra? Sketch the graph or build a quick table to sanity-check.
• Read $L$ off a graph? Verify with substitution if a formula is available.
• Choose the most informative representation for the question — sometimes a picture is instant, sometimes algebra is exact.

Write from four views that disagree

• Algebra says a limit is 4.
• A quick graph was read as 5.