Limits from Graphs and Tables

≈ 25 min

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Limits from Graphs and Tables

A limit records the value approached from both sides. One-sided limits must agree for a two-sided limit to exist; the actual function value may be different or absent.

Worked reasoning

A two-sided limit requires equal one-sided limits.
Because $5\ne3$ , the two-sided limit does not exist.

Key idea

A two-sided limit exists only when left- and right-hand limits agree.

If $\lim_{x\to2^-}f(x)=5$ and $\lim_{x\to2^+}f(x)=3$ , what is $\lim_{x\to2}f(x)$ ?

It does not exist

Which statement best captures the central mathematical idea in Limits from Graphs and Tables?

A two-sided limit exists only when left- and right-hand limits agree.

The limit must equal the displayed function value at the point.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Limits from Graphs and Tables, which move is most reliable?

Trace the function toward the target from each side separately.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Limits from Graphs and Tables?

The limit must equal the displayed function value at the point.

A two-sided limit exists only when left- and right-hand limits agree.

Trace the function toward the target from each side separately.

Compare both one-sided approaches and distinguish them from $f(a)$ .

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