Is the function $\frac{\sin{x}}{x}$ continuous at $x=0$?

Question

Here the function is $$f(x) =\frac{\sin{x}}{x}.$$ We see that the right hand limit equals the left hand limit but does $f(0)$ exist?

Answer 1

Indeed, the given function $$f(x) := \frac{\sin{x}}{x}$$ is not defined at $x=0$. Because $f(0)$ isn't defined (at this stage), it does not really make sense to ask if $f$ is continuous at the point $x=0$. This is because continuity at $x=0$ requires that $$ \lim_{x \to 0} f(x) = f(0). $$ However, this is meaningless if we don't know what $f(0)$ is!

Now, as you've observed, the right and left limits agree, i.e. $$ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^-} f(x) = 1. $$ This means that the function $f$ can be made continuous at $x=0$ by assigning it the value $f(0) := 1$. Alternatively, this means that the function $$ f(x) := \begin{cases} \frac{\sin{x}}{x} & \text{if }x \neq 0,\\ 1 & \text{if }x = 0 \end{cases} $$ is continuous everywhere.

Answer 2

As it is $f(0)$ is not defined and $f$ is continuous on $\mathbb R \setminus \{0\}$. If you extend the definition of $f$ by defining $f(0)$ as $1$ then it becomes continuous.

Answer 3

$f(0)$ is undefined since $\sin(0)/0$ is undefined. However, the limit of $f(x)$ as $x$ approaches zero, as you noted, is one. If you define $f(0)=1$, then $f$ is continuous (everywhere) due to this.