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Indeterminate forms in limits of a sequances

Limit of a sequance Lecture 3

Topic: Indeterminate forms


In the article I will present what are indeterminate forms appearing in tasks on the limit of the sequence.

Easy to define limits of sequences

How to calculate the limits of a sequence? If you came across this topic in college or elsewhere, you certainly associate it with “methods”, “parenthesis”, “multiplication by conjugate”, etc. And rightly so. But many sequences have such a simple limit to compute that using some complex methods is a waste of time and effort at best.

Example 1

Let’s analyze the sequence:


The next words in the sequence would look like this:

\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6}, ... ,\frac{1}{100}, ... ,\frac{1}{1000}, ... etc.

We see that its numbers are getting smaller and smaller and converges to zero. After thinking about it, you can come to the conclusion that a similar result of the limit will be obtained in every situation in which the numerator converges to a constant number and the denominator diverges to infinity. The limit will then always be zero – because with larger and larger denominators, the whole expression will be smaller and smaller – converging to zero. Therefore:

[ \frac{A}{\infty} ] =0

…regardless of what specific sequence is at the bottom of the denominator, as long as it goes to infinity.

When will it look different?

Indeterminate form of type \frac{\infty}{\infty}

Example 2

Let’s take two sequences: {a}_{n}={n}^{2}-1 and {b}_{n}={n}^{2}+1 . Both diverge to infinity. If we divide their corresponding terms, we get a new sequence:


Symbolically, such a situation is \frac{\infty}{\infty} – this is a designation for a sequence consisting of dividing two others diverging to infinity. What will be its limit?

Writing \frac{{n}^{2}-1}{{n}^{2}+1} we get:

0,\frac{3}{5},\frac{8}{10},\frac{15}{17},\frac{24}{26},\frac{35}{37},\frac{48}{50},\frac{63}{65},\frac{80}{82},\frac{99}{101}, ...

These numbers are getting closer and closer to one. For this particular sequence so: \frac{\infty}{\infty} \rightarrow 1

Example 3
Let’s take two other sequences: {a}_{n}=4n+1 and {b}_{n}={n}^{2} . Both diverges to infinity. By dividing their terms, we get the following sequence:


This is a sequence in which, again, in the numerator and denominator we have sequences diverging to infinity, so again the situation \frac{\infty}{\infty} . But what will be its limit this time?

Writing down the successive terms of the sequence \frac{4n+1}{n^2} we get:

5,2 \frac{1}{4} ,1 \frac{4}{9} ,1\frac{1}{16} , \frac{21}{25}, \frac{25}{36}, \frac{29}{49}, \frac{33}{64}, \frac{37}{81} , \frac{41}{100}, ...

So you can see that the denominator goes “faster” to infinity than the numerator, and the whole expression converges to 0.

In both examples (2 and 3) we had the same situation: \frac{\infty}{\infty} and two different results: 1 and 0. It is not difficult to imagine various other possibilities, for example, the numerator running “faster” than the denominator – then the sequence will run to infinity.

In a \frac{\infty}{\infty} situation we are not able to determine at the very beginning what the limit of the sequence is and we have to use various transformations of the expression from which we calculate the limit.

Expressions not generally marked

We call these types of expressions “indeterminate forms” (there are seven of them in total). Counting the limit of a sequence from indeterminate symbols requires the use of various methods, but if there are no undetermined expressions in the sequence – it is usually a sequence whose limit can be determined very easily.

So let’s list all seven unsigned symbols:

[ \frac{\infty}{\infty} ],[ \frac{0}{0} ],[ 0 \cdot \infty ],[ \infty-\infty ],[ {1}^{\infty} ],[ {0}^{0} ],[ {\infty}^{0} ]

The important thing is to understand that an unmarked expression is a certain symbol that carries meanings different from those to which we may have become accustomed. For example, the symbol 0 in unmarked expressions does NOT mean the number zero (as many people mistakenly believe…) but a sequence with a limit equal to zero – and that’s something else entirely, right? In symbol [ \frac{0}{0} ] so we don’t have “division by zero”, only the quotient of two sequences approaching zero.

Click to recall how to count the limits of a sequence from definitions (previous Lecture)< —

Click to see the definition of the limit of an improper sequence (next Lecture) –>

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