So the sequence is just $1,1,1,1, \dots$, and so the limit is $1$ (and not $0$!). But this time when we evaluate the function at these points, we get $\sin(2n\pi + \pi/2) = \sin(\pi/2) = 1$. (1) Each sequence means that, no matter what sequence $\left(x_n\right)_, \dots $$Īgain, this sequence also converges to $0$. An arithmetic sequence is defined in two ways.It is a 'sequence where the differences between every two successive terms are the same' (or) In an arithmetic sequence, 'every term is obtained by adding a fixed number (positive or negative or zero) to its previous term'. How does this then fit with the definition? But - $f(0)$ is not even defined, since that'd be a division by $0$. Lets discuss these ways of defining sequences in more detail, and take a look at some examples. When I look at a graph of for example the topologist's sine function, and we can focus on the interval $(-1, 1)$ and we let $a=0$ - then I intuitively think about two sequences on the $x$-axis: the sequence of real numbers approaching $0$ from the left and one approaching $0$ from the right (just "following" the $x$-axis in my mind towards $0$.) And, intuitively, I want to verify that both tend or not tend to $f(a)$ according to the definition.What does "each sequence" (in the interval $A$) actually mean? My understanding is that we want to ensure there are no jumps/gaps in the graph of $f$, thus we'd need to check the output of $f$ for every real number on the $x$-axis to give a point on the $y$-axis - which is of course not possible - but how does "every sequence in $A$" cover all real numbers in the interval on the $x$-axis?.if for each sequence $(x_n)$ in $A$ such that $x_n \to a$. I've watched some videos on this and also read other questions, but I'm not fully onboard with this definition yet. After she dropped the photographs, they were out of sequence. The sequential definition is the following:Ī function $f: A \to \mathbb R$ is continuous at a point $a \in A$ if for each sequence $(x_n)$ in $A$ such that $x_n \to a$, we have $f(x_n) \to f(a)$. Britannica Dictionary definition of SEQUENCE 1 : the order in which things happen or should happen count a sequence of events noncount He listened to the telephone messages in sequence. I've been exposed to both the classic and sequential definition of continuity.
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