Year 11: Understanding limits

Limits are the bridge into calculus. They let you describe what a function is doing as $x$ gets close to a value, even if direct substitution is awkward or impossible. In Year 11, limits matter because they explain why derivatives work, why reciprocal graphs have vertical behaviour near excluded values, and how we handle rational expressions that simplify only after factorisation.

A limit asks what value

f (x)

is approaching as

x

approaches a number. It is about the trend near the point, not just the function value exactly at the point.

What to master

Using direct substitution when the function is continuous at the target value.
Recognising when a rational expression gives an indeterminate form like $0 / 0$ .
Factorising and simplifying before taking the limit.
Interpreting one-sided behaviour for reciprocal-style graphs.

Direct substitution

For polynomials and many simple expressions, the quickest method is direct substitution. If the function has no denominator issues, root restrictions, or other discontinuities, plug the target value straight in.

Example 1: Evaluate

lim_{x \to 3} (2 x^{2} - x + 1)

Substitute $x = 3$ .
$2 {(3)}^{2} - 3 + 1 = 18 - 3 + 1$ .
$= 16$ .

Answer: the limit is

16

This works because polynomials are continuous. Near $x = 3$ , the outputs stay close to the value at $x = 3$ , so the limit and the function value match.

Rational limits and the $0 / 0$ issue

If substitution gives $0 / 0$ , that does not mean the limit is zero. It means the expression needs more work. Often a common factor can be cancelled first.

Example 2: Evaluate

lim_{x \to 2} (x^{2} - 4) / (x - 2)

Direct substitution gives $(4 - 4) / (2 - 2) = 0 / 0$ .
Factor the numerator: $x^{2} - 4 = (x - 2) (x + 2)$ .
So the expression becomes $((x - 2) (x + 2)) / (x - 2)$ .
Cancel the common factor to get $x + 2$ , for $x \neq 2$ .
Now take the limit: $lim_{x \to 2} (x + 2) = 4$ .

Answer: the limit is

4

This is a standard Year 11 pattern. The original expression is undefined at the target point, but the nearby values still approach a definite number.

One-sided limits

Reciprocal graphs show why the direction of approach matters. For $y = 1 / x$ , values behave very differently on the left and right of $x = 0$ .

Example 3: Describe the behaviour of

1 / x

x

approaches

0

If $x$ approaches $0$ from the positive side, $1 / x$ becomes very large positive.
If $x$ approaches $0$ from the negative side, $1 / x$ becomes very large negative.

Because the left-hand and right-hand behaviour do not match, the two-sided limit at

x = 0

does not exist.

This is closely related to vertical asymptotes. A function can get arbitrarily large in magnitude near an x-value without ever settling to one finite limit.

Why limits matter for derivatives

When you calculate a derivative from first principles, you start with an average rate of change over a tiny interval, then let that interval shrink. That "shrinking" process is a limit. So even if Year 11 derivatives are taught mostly through rules and interpretation, limits are the logic underneath.

Example 4: Consider

(f (2 + h) - f (2)) / h

h \to 0

This expression measures the average rate of change over a smaller and smaller interval. The limit tells you the instantaneous rate of change at $x = 2$ , which is the derivative there.

Common mistakes

Treating $0 / 0$ as a final answer instead of a signal to simplify.
Cancelling terms instead of cancelling factors.
Ignoring one-sided behaviour for reciprocal or rational functions.
Confusing the value of a function at a point with the limit near that point.

Study routine

A good limit routine is: substitute first, classify the result, then simplify only if needed. If the expression is rational, look for factorisation before anything else. If the graph is reciprocal in style, check left-hand and right-hand behaviour separately.