Calculus I · Limits and Continuity · lesson

Function Value vs. Limit

Concept

Learning objectives

Determine $f(a)$ and $\lim_{x\to a}f(x)$ separately; explain why changing one isolated function value does not change the limit.

The Function Value and the Limit Are Different Questions

The expression $f(a)$ asks:

What output has the function assigned at the single input $a$ ?

The expression

\lim_{x\to a}f(x)

asks:

What value do nearby outputs approach when nearby inputs approach $a$ ?

These answers can agree, disagree, or one may fail to exist.

Guided walkthrough

The dot and the road

Imagine a road drawn toward the point $(2,5)$ , but someone places a separate dot at $(2,9)$ . The road tells you where nearby points are going. The separate dot tells you the actual value at $x=2$ .

Thus,

\lim_{x\to2}f(x)=5, \qquad f(2)=9.

The limit follows the road, not the misplaced dot.

Interactive checklimit-hole-01

Evaluate $\lim_{x\to2}\frac{x^2-4}{x-2}$ .

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Factor the difference of squares.

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Worked example

A piecewise value does not control the limit

Let

g(x)= \begin{cases} x^2+1,&x\ne2,\\ 9,&x=2. \end{cases}

Find $g(2)$ and $\lim_{x\to2}g(x)$ .

Show worked solution

The piecewise definition directly states

\boxed{g(2)=9}.

For all nearby inputs $x\ne2$ , the rule is $g(x)=x^2+1$ . Therefore,

\begin{aligned} \lim_{x\to2}g(x) &=\lim_{x\to2}(x^2+1)\\ &=2^2+1\\ &=\boxed{5}. \end{aligned}

g(2)\ne\lim_{x\to2}g(x).

Parabola with an open point at (2, 5) and a filled point at (2, 9). — **Read the graph**
The nearby parabola determines the limit; the filled point determines the function value.

Summary

A single point has no control over a two-sided limit. You may change $f(a)$ , delete it, or define it later, and the limit remains the same as long as all nearby values with $x\ne a$ remain unchanged.

Quick check

Suppose $f(x)=3x-1$ for every $x\ne4$ , while $f(4)=-100$ . Find

f(4) \qquad\text{and}\qquad \lim_{x\to4}f(x).

Answer. $f(4)=-100$ , but

\lim_{x\to4}f(x)=3(4)-1=11.

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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