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#3 Re: Help Me ! » Limit of Two-piece Function » 2021-06-05 22:48:01

Your conclusion is correct, but the same comments I made on your other piecewise function thread also apply here so I suggest you read that first (it's the same kind of set-up, just with two pieces rather than three).

The question did ask for you to use a graph. So what do you think the graph looks like and in particular what do you think it looks like near x = -1?

#4 Re: Help Me ! » Limit of Three-piece Function » 2021-06-05 22:42:33

Your conclusion is correct -- however, there are a few technical points to note.

nycmathguy wrote:

Find the limit of x^2 as x tends to 1 from the left side.

What you really want to be saying is that you are finding the limit of f(x) as x tends to 1 from the left, rather than the limit of x^2.

nycmathguy wrote:

(1)^2 = 1

Strictly speaking, this is how you evaluate the limit of f(x) as x approaches 1 from the left, but this isn't how you evaluate the left-hand limit of x^2, this is just substituting in x = 1 -- although you can get away with it here. (This is because x^2, 2 and -3x + 2 are all continuous functions, so the left and right-hand limits for each function exist, are equal, and are equivalent to their usual limits.) The way you ought to be thinking about it is:

It'll lead to the same answer, but conceptually this is along the sort of lines you'll be required to think through when you look at the epsilon-delta type questions.

However, the question did ask for you to use a graph. So what do you think f(x) might look like on a graph? What kind of shape does it have as you move from left to right? And in particular what happens near x = 1?

#23 Re: Help Me ! » Repeating numbers » 2021-05-21 08:42:12

This looks like the decimal representation of
, which also explains why you don't get the same sort of pattern when you multiply your number by 7, since you'd get
which has a different string of recurring decimal places.

Since
is a rational number, then its decimal expansion will recur (i.e. repeat). The length of the string of digits which recurs is usually called the 'period', and the string of digits which repeats itself is sometimes called the 'repetend'. If you multiply this by any number which is coprime to 49, then you'll end up with the same sequence of decimal places, but 'shifted along' by some number.

Bob's example is a good one because 1/7 is a cyclic number, i.e. if you calculate successive multiples of 1/7 (i.e. 2/7, 3/7, 4/7, etc...) then you're shifting the decimal places by some amount each time (but preserving their 'order'). In other words, you get cyclic permutations. 1/7 has a recurring decimal expansion with period 6.

You can sometimes use properties of periods to determine what the length of the period is (or at least reduce the number of possibilities to a more manageable size!). For example, it's true that if a is coprime to b (that means that a and b share no common factors apart from 1), then the period of a/b is the same as the period of 1/b. This means, for example, that the period of 100/49 is the same as the period of 1/49. But since 49 is the power of a prime (it's 7 x 7 = 49), then since:

then you can say that

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