Power series, Fourier series and solutions of differential equations are all limits of sequences of functions. The simplest way to take such a limit is one point at a time, using what you know about limits of sequences. It is simple, and it is too weak to keep continuity.
🔑 Key idea
$f_n \to f$ pointwise on $S$ when $f_n(x) \to f(x)$ for each $x \in S$ separately: for every $x$ and every $\varepsilon > 0$ there is an $N$ (which may depend on $x$) with $|f_n(x) - f(x)| < \varepsilon$ for $n \ge N$. A pointwise limit of continuous functions can be discontinuous.
💡 Why it works
Fix $x$, and $(f_n(x))$ is an ordinary sequence of numbers. For $f_n(x) = x^{n}$ on $[0, 1]$: if $0 \le x < 1$ then $x^{n} \to 0$, and at $x = 1$ every term is $1$. So the limit is $0$ on $[0, 1)$ and $1$ at $1$: it jumps, although every $x^{n}$ is continuous. The reason is visible in the $N$: to get $x^{n} < \frac{1}{2}$ you need $n > \frac{\ln 2}{-\ln x}$, which grows without bound as $x \to 1$. Points near $1$ converge more and more slowly, and no single $N$ serves them all.
In symbols, continuity of the limit would mean swapping two limits: $\lim_{x \to 1}\lim_{n \to \infty} x^{n} = 0$, while $\lim_{n \to \infty}\lim_{x \to 1} x^{n} = 1$. Limits cannot always be swapped, and the rest of this unit is about when they can.
$f_n(x) = x^{n}$ on $[0, 1]$ for growing $n$ (earlier ones faint): each graph is continuous, but near $x = 1$ they bend up later and later, and the limit jumps at $1$.
Worked examples
✏️ Worked example: A limit with three cases
Find the pointwise limit of $f_n(x) = \frac{x^{n}}{1 + x^{n}}$ on $[0, \infty)$.
This is the classic limit $\left(1 + \frac{x}{n}\right)^{n} \to e^{x}$.
Answer: $e^{2}$. Here the limit function $e^{x}$ is continuous.
✏️ Worked example: A special point
Find the pointwise limit of $f_n(x) = \arctan(nx)$ at $x = 0$ and at $x = \frac{1}{100}$.
At $0$ every term is $0$.
At $\frac{1}{100}$, $nx \to \infty$, so $\arctan(nx) \to \frac{\pi}{2}$.
Answer: $0$ and $\frac{\pi}{2}$: the limit jumps at $0$.
⚠️ Common mistake
Missing the special points. Students compute the limit for a "typical" $x$ and forget the boundary cases: $x = 1$ for $x^{n}$, $x = 0$ for $\arctan(nx)$ or $\frac{nx}{1 + nx}$. Before writing down $f$, list the values of $x$ where the behavior of $f_n(x)$ changes (where a base equals $1$, or a product $nx$ stays $0$), and treat each separately.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 2 XP.