Limit of a general rational function

Brief explanation: Today we continue the study of limits, this time not with an epsilon delta proof but with a general limit that I believe will get us to understand some important facts about polynomials and their division.

(This exercise was taken from Michael Spivak’s Calculus)

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Question: Calculate the following limit:

enunciado.png

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Comprehending:

(If you substitute the x by y directly you’ll get immediately an indetermination 0/0, so I didn’t even consider doing that)

First of all, we need to understand what the expression inside the limit means. Let’s look at the numerator first. The way I normally use to understand these kinds of problems is to substitute the n in the expression by an integer and see where it goes. Let’s use n = 2:

1.png

Now, n = 3:

2.png

The n = 4 will make this even clearer:

3.png

Do you see a pattern emerging here? We can always divide our polynomial by (x – y)! Now let’s generalize a formula for the nth case:

4.png

Compare this formula with the n = 4 case and you’ll understand it. I’ll prove this formula quickly. If we multiply the (x – y) in the above expression:

5.png

This is equal to:

6.png

Notice that most terms will cancel out! In the end, we get (x^n – y^n).

Now all we need is to apply this logic to the limit.

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Solving: We now know that:

7.png

The terms (x – y) just cancel out:

8.png

Taking the limit, we get the following:

9.png

How many y^n-1 do we have? Look at the n = 4 case. We have, after (x – y),

3

Five terms! Since n = 4, we have n + 1 terms. Now, our maximum term was n – 1, so we’ll get (n – 1) + 1 = n terms!

(If this step confused you, try it for many different n cases. If it still confuses you, tell me!)

So, we get:

10.png

So, our final answer is:

11.png

And we’re done!

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