Basic Limits and Continuity

students, calculus begins with a simple but powerful question: what happens to a function as its input gets closer and closer to a value? 🚀 That question leads to limits, and from limits we build continuity, which helps us understand whether a graph has a break, a gap, or a smooth path. These ideas are essential in IB Mathematics: Analysis and Approaches HL because they support differentiation, integration, and many real-world models.

Objectives

By the end of this lesson, you should be able to:

explain the meaning of a limit using clear mathematical language,
find basic limits from graphs, tables, and algebraic rules,
determine whether a function is continuous at a point,
connect limits and continuity to later calculus ideas such as derivatives and integrals,
use examples from functions and graphs to justify your reasoning.

Think of limits like approaching a destination on a map 🗺️. You may not be standing exactly on the destination yet, but you can still describe where you are heading. Continuity asks whether the route is smooth or whether there is a missing bridge or a jump.

What a Limit Means

A limit describes the value that a function approaches as the input approaches a certain number. If $f(x)$ gets closer to some value $L$ as $x$ gets closer to $a$, we write

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

This does not always mean that $f(a)=L$. That is one of the most important ideas in the topic. The limit is about nearby values, not only the exact point.

For example, consider $f(x)=2x+1$. As $x$ gets close to $3$, the output gets close to $7$. So

$\lim_{x \to 3} (2x+1)=7$

This is easy because linear functions are smooth and predictable. A student may also check it by substitution when the function is continuous. In fact, for many basic functions, the limit can be found by direct substitution.

Limits can also be one-sided. A left-hand limit is written

$\lim_{x \to a^-} f(x)$

and a right-hand limit is written

$\lim_{x \to a^+} f(x)$

These matter when a function behaves differently on each side of a point. For a two-sided limit to exist, the left-hand and right-hand limits must be equal.

Example: A piecewise function

Let

$f(x)=\begin{cases}$

2x, & x<1 \\

$5, & x=1 \\$

3x-1, & x>1

$\end{cases}$

To study the limit as $x \to 1$, look at each side:

from the left, $f(x)=2x$, so $\lim_{x \to 1^-} f(x)=2$,
from the right, $f(x)=3x-1$, so $\lim_{x \to 1^+} f(x)=2$.

Since both one-sided limits are equal,

$\lim_{x \to 1} f(x)=2$

Notice that $f(1)=5$, which is different from the limit. This shows that a limit can exist even when the actual function value is not the same as the limit.

How to Find Basic Limits

There are several common ways to evaluate basic limits in IB Mathematics: Analysis and Approaches HL.

1. Direct substitution

If the function is well behaved at the point, substitute the value of $x$ directly.

Example:

$\lim_{x \to 4} \sqrt{x+5} = \sqrt{9}=3$

This works because square root functions are continuous where they are defined.

2. Simplifying algebraically

Sometimes direct substitution gives an indeterminate form like $\frac{0}{0}$. That does not mean the limit does not exist; it means more work is needed.

Example:

$\lim_{x \to 2} \frac{x^2-4}{x-2}$

Factor the numerator:

$\frac{x^2-4}{x-2}$=$\frac{(x-2)(x+2)}{x-2}$=x+2 \quad \text{for } x\ne 2

Now substitute:

$\lim_{x \to 2} \frac{x^2-4}{x-2}=4$

Even though the original expression is undefined at $x=2$, the limit still exists.

3. Using graphs and tables

A graph gives visual evidence. If the $y$-values approach the same number from both sides, the limit exists. A table can show the same pattern numerically.

For example, if values of $x$ near $1$ produce outputs near $3$, then it is reasonable to conclude that

$\lim_{x \to 1} f(x)=3$

This kind of reasoning is important in IB because not every limit can be solved only by substitution.

4. Recognizing when a limit does not exist

A limit may fail to exist if the left-hand and right-hand limits are different, or if the function grows without bound near the point.

Example:

$\lim_{x \to 0} \frac{1}{x}$

As $x \to 0^+$, the expression becomes very large positive, and as $x \to 0^-$, it becomes very large negative. Since the one-sided limits are not equal, the two-sided limit does not exist.

Continuity and What It Means

A function is continuous at a point if its graph has no break, hole, or jump there. In formal terms, a function $f$ is continuous at $x=a$ if all three conditions hold:

$f(a)$ is defined,
$\lim_{x \to a} f(x)$ exists,
$\lim_{x \to a} f(x)=f(a)$.

These conditions connect the idea of a limit with the actual point on the graph.

Example: Continuous at a point

Let $f(x)=x^2$. Check continuity at $x=2$:

$f(2)=4$,
$\lim_{x \to 2} x^2=4$,
the limit equals the function value.

So $f$ is continuous at $x=2$.

Example: Not continuous at a point

Consider

$g(x)=\begin{cases}$

$\frac{x^2-1}{x-1}, & x\ne 1 \\$

$0, & x=1$

$\end{cases}$

For $x\ne 1$, simplify:

$\frac{x^2-1}{x-1}=\frac{(x-1)(x+1)}{x-1}=x+1$

$\lim_{x \to 1} g(x)=2$

But $g(1)=0$. Since the limit and function value are different, $g$ is not continuous at $x=1$. The graph would have a hole at the point $(1,2)$ and a filled dot at $(1,0)$.

Types of discontinuity

Common discontinuities include:

removable discontinuity: a hole that could be fixed by redefining the function value,
jump discontinuity: the graph jumps from one height to another,
infinite discontinuity: the function grows without bound near a vertical asymptote.

These names help describe what kind of break appears on the graph.

Why Limits and Continuity Matter in Calculus

Limits are the foundation of derivatives and integrals. A derivative is built from a limit of difference quotients:

$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$

Without limits, there would be no precise way to define instantaneous rate of change. That is why limits are central to motion, growth, and optimisation problems.

Continuity also matters because many theorems in calculus require it. For example, if a function is continuous on a closed interval, then it has useful properties such as taking every value between its endpoints. This helps when solving equations numerically or analyzing physical models.

In real life, continuity is often a good approximation. A car moving along a road usually changes position continuously, while a graph of ticket prices might jump at certain thresholds. In both cases, limits help describe what is happening near important points.

Worked IB-Style Reasoning

Suppose a function models the temperature in a room:

$T(t)=\begin{cases}$

$18+2t, & 0\le t<3 \\$

$25, & t=3 \\$

24-t, & t>3

$\end{cases}$

To check continuity at $t=3$:

left-hand limit: $\lim_{t \to 3^-} T(t)=18+2(3)=24$,
right-hand limit: $\lim_{t \to 3^+} T(t)=24-3=21$.

Because the one-sided limits are not equal, $\lim_{t \to 3} T(t)$ does not exist. So the function is not continuous at $t=3$.

This shows the exact reasoning expected in IB: identify the relevant expressions, calculate carefully, and state the conclusion clearly with mathematical justification.

Conclusion

students, basic limits and continuity are the gateway to calculus 🌟 A limit describes what a function is approaching, while continuity checks whether that approach matches the actual function value. These ideas help you analyze graphs, solve equations, and understand the foundations of differentiation and integration. In IB Mathematics: Analysis and Approaches HL, strong limit and continuity reasoning supports everything that comes later in the calculus topic.

Study Notes

A limit describes the value a function approaches as $x$ approaches a point $a$.
The notation $\lim_{x \to a} f(x)=L$ means the outputs of $f(x)$ get close to $L$ near $a$.
A two-sided limit exists only if the left-hand limit and right-hand limit are equal.
Direct substitution works when the function is continuous at the point.
If substitution gives $\frac{0}{0}$, simplify algebraically before evaluating the limit.
A function is continuous at $x=a$ if $f(a)$ exists, $\lim_{x \to a} f(x)$ exists, and both are equal.
Common discontinuities are removable, jump, and infinite discontinuities.
Limits are the basis of the derivative formula $f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}$.
Continuity is important in modelling because many real-world systems change smoothly.
In IB exam questions, always show clear steps and state whether the limit exists and whether the function is continuous.