What Is a Limit and How Do You Estimate One From a Table or Graph?

A limit is the value that a function approaches as the input approaches a particular value. To estimate one, examine function values or graph behaviour from both the left and the right of that input.

This is SL 5.1: Introduction to Limits in IB Mathematics: Applications and Interpretation. The notation

$$\lim_{x \to a} f(x)=L$$

means that $f(x)$ approaches $L$ as $x$ approaches $a$. It does not necessarily mean that $x=a$ or that $$.

Estimating From a Table

Use values of $x$ increasingly close to $a$, including values on both sides. For example:

As $x$ approaches $2$ from below, $f(x)$ approaches $4$. The same happens as $x$ approaches $2$ from above. Therefore,

$$\lim_{x \to 2} f(x)=4.$$

Choose table entries close enough to reveal the trend, rather than values far from $a$. If the outputs stabilize near one number on both sides, that number is the estimate. Keep several decimal places while inspecting the pattern, because premature rounding can hide whether the two sides agree.

Estimating From a Graph

Trace the curve toward $x=a$ from each side and identify the corresponding $y$-value approached. For a graph, distinguish an open circle showing a missing function value from the nearby curve’s approached height.

A common misconception is that the limit must equal $f(a)$. This is false: a graph may have a hole at $x=a$, or $f(a)$ may be defined at a different height, while the limit still exists.

Exam Technique

In an SL Paper 1 or Paper 2 response, state the value approached from both sides and then give the limit using correct notation. SL 5.1 tests graphical or numerical estimation; formal algebraic limit methods are not required. Do not rely on a single table value or simply substitute $x=a$.