Limits with Piecewise Functions: Quick Review

Concept: Limits vs Point Value

• A limit describes f(x) as x approaches a, not the actual value at a.
• The limit can exist even if f(a) is undefined or f(a) differs from the limit value.
• If the left-hand and right-hand limits exist and are equal, the (two-sided) limit equals that value.

Piecewise Definition at the Critical Point

• Typical setup: $f(x) = \begin{cases} g(x), & x \ne a \ h, & x = a \end{cases}$
• Then: $\lim<em>{x \to a} f(x) = \lim</em>{x \to a} g(x)$ provided the limit exists.
• The value h at x = a does not affect the limit (only the value of g near a does).

Graphical Interpretation

• There is a hole at x = a with y = L (the limit value) on the graph of the non-defective piece.
• There may be a separate filled dot at (a, f(a)) = (a, h) indicating the actual function value at a.
• Consequently, f(a) may differ from lim_{x\to a} f(x).

Transcript-inspired Example (a = 2)

• If $\lim_{x \to 2} f(x) = 2$ but $f(2) = -1$, the graph has an open circle at (2, 2) and a filled dot at (2, -1).
• The left-hand and right-hand limits both approach 2.

Quick Template for Exam

• For a point a, define: $f(x) = \begin{cases} g(x), &amp; x \ne a \\ h, &amp; x = a \end{cases}$
• Then: $\lim<em>{x \to a} f(x) = \lim</em>{x \to a} g(x)$ (if this limit exists).
• If $\lim_{x \to a} g(x) = L$, then the limit of f at a is L even if f(a) = k \neq L.