It's been many years since I've done math so I've decided to take a Calculus refresher course on edx, there's the following problem in the limits section:
I'm trying to understand why $\lim_{x \to 1}f(x)$ is 1 but $\lim_{x \to 1}f(1)$ is 2. We're plugging 1 in both cases, why are the results different? Is it because in the former case we're evaluating a variable and in the latter a constant?
Similarly, $\lim_{x \to 4}f(x)$ is 2 (I'm guessing it's the same reason $\lim_{x \to 1}f(x)$ is 1).
$\endgroup$ 01 Answer
$\begingroup$According to the graph, $f$ is not continuous in $1$. You see indeed that $f(1) = 2$, but that when $x$ approaches $1$ (by the left or by the right), the value of $f(x)$ approaches $1$.
This means that $\lim_{x \rightarrow 1} = 1$. But $f(1)=2$, which is not contradictory since $f$ is not continuous in $1$.
For the same reason, when $x$ approaches $4$ (but is not equal to $4$), you see that the value of $f(x)$ approaches $2$. So $\lim_{x \rightarrow 4} = 2$, and that is independant of the value of $f(4)$ (which you don't know by the way).
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