# 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..... 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={1,3,5,7,9,...} and sequence y_n={2,4,6,8,10,...} then x_n+y_n={1+2,3+4,5+6,7+8,9+10,...}={3,7,11,15,19,...}.

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 (x_n+y_n)=a+b, lim(x_n-y_n)=a-b.

This fact holds for any finite number of summands.

Example 1 . Let x_n=1/n, y_n=1+1/n^2, z_n=1/n^3.

Then

lim (x_n+y_n-z_n)=lim x_n+lim y_n-lim z_n=lim 1/n+lim (1+1/n^2)-lim 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 (x_ny_n)=ab.

This fact holds for any finite number of factors.

Example 2. Let x_n=1/n, y_n=1/n.

Then lim 1/n^2=lim (x_n y_n)=lim(1/n*1/n)=lim 1/n * lim 1/n=0*0=0.

So, lim 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 (b!=0) then their quotient also has finite limit and lim (x_n)/(y_n)=a/b.

Example 3. Let x_n=1/n, y_n=1+1/n.



Then lim (x_n)/(y_n)=lim (1/n)/(1+1/n)=(lim 1/n) / (lim (1+1/n))=0/1=0.