Algebra of Limit of Sequence

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Consider two sequences $$${x}_{{n}}$$$ and $$${y}_{{n}}$$$. When we talk about sum of these sequences, we talk about sequence $$${x}_{{n}}+{y}_{{n}}$$$, whose elements are $$${x}_{{1}}+{y}_{{1}},{x}_{{2}}+{y}_{{2}},{x}_{{3}}+{y}_{{3}}\ldots.$$$. Same can be said about other arithmetic operations. In other words sum of sequences is sequence with elements that are sum of corresponding elements of initial two sequences.

For example, consider sequence $$${x}_{{n}}={\left\{{1},{3},{5},{7},{9},\ldots\right\}}$$$ and sequence $$${y}_{{n}}={\left\{{2},{4},{6},{8},{10},\ldots\right\}}$$$ then $$${x}_{{n}}+{y}_{{n}}={\left\{{1}+{2},{3}+{4},{5}+{6},{7}+{8},{9}+{10},\ldots\right\}}={\left\{{3},{7},{11},{15},{19},\ldots\right\}}$$$.

Following facts are important because with their help in many cases we can easily find limit without using definition.

Limit of sum (difference) equals sum (difference) of limits. If sequences $$${x}_{{n}}$$$ and $$${y}_{{n}}$$$ have finite limits: $$$\lim{x}_{{n}}={a}$$$ and $$$\lim{y}_{{n}}={b}$$$, then their sum and difference also have finite limits, and $$$\lim{\left({x}_{{n}}+{y}_{{n}}\right)}={a}+{b}$$$, $$$\lim{\left({x}_{{n}}-{y}_{{n}}\right)}={a}-{b}$$$.

This fact holds for any finite number of summands.

Example 1. Let $$${x}_{{n}}=\frac{{1}}{{n}}$$$, $$${y}_{{n}}={1}+\frac{{1}}{{{n}}^{{2}}}$$$, $$${z}_{{n}}=\frac{{1}}{{{n}}^{{3}}}$$$.

Then

$$$\lim{\left({x}_{{n}}+{y}_{{n}}-{z}_{{n}}\right)}=\lim{x}_{{n}}+\lim{y}_{{n}}-\lim{z}_{{n}}=\lim\frac{{1}}{{n}}+\lim{\left({1}+\frac{{1}}{{{n}}^{{2}}}\right)}-\lim\frac{{1}}{{{n}}^{{3}}}=$$$

$$$={0}+{1}-{0}={1}$$$.

Limit of product equals product of limits. If sequences $$${x}_{{n}}$$$ and $$${y}_{{n}}$$$ have finite limits: $$$\lim{x}_{{n}}={a}$$$ and $$$\lim{y}_{{n}}={b}$$$, then their product also has finite limit and $$$\lim{\left({x}_{{n}}{y}_{{n}}\right)}={a}{b}$$$.

This fact holds for any finite number of factors.

Example 2. Let $$${x}_{{n}}=\frac{{1}}{{n}}$$$, $$${y}_{{n}}=\frac{{1}}{{n}}$$$.

Then $$$\lim\frac{{1}}{{{n}}^{{2}}}=\lim{\left({x}_{{n}}{y}_{{n}}\right)}=\lim{\left(\frac{{1}}{{n}}\cdot\frac{{1}}{{n}}\right)}=\lim\frac{{1}}{{n}}\cdot\lim\frac{{1}}{{n}}={0}\cdot{0}={0}$$$.

So, $$$\lim\frac{{1}}{{{n}}^{{2}}}={0}$$$.

Limit of quotient equals quotient of limits. If sequences $$${x}_{{n}}$$$ and $$${y}_{{n}}$$$ have finite limits: $$$\lim{x}_{{n}}={a}$$$ and $$$\lim{y}_{{n}}={b}$$$ $$$\left({b}\ne{0}\right)$$$ then their quotient also has finite limit and $$$\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\frac{{a}}{{b}}$$$.

Example 3. Let $$${x}_{{n}}=\frac{{1}}{{n}}$$$, $$${y}_{{n}}={1}+\frac{{1}}{{n}}$$$.

Then $$$\lim\frac{{{x}_{{n}}}}{{{y}_{{n}}}}=\lim\frac{{\frac{{1}}{{n}}}}{{{1}+\frac{{1}}{{n}}}}=\frac{{\lim\frac{{1}}{{n}}}}{{\lim{\left({1}+\frac{{1}}{{n}}\right)}}}=\frac{{0}}{{1}}={0}$$$.