Limits part 1: Infinite sequences: Basic calculation rules for limits
Calculation rules for limits
As you discovered in the previous exercise, you can calculate the limit of a sum if you already know the limits of the two separate terms:
Sum rule If \(\lim_{n\to\infty} a_n = L\) and \(\lim_{n\to\infty} b_n = M\) then we have \[ \lim_{n\to\infty} \left( a_n + b_n\right) = L+M\text. \]
There is also a product rule
Product rule If \(\lim_{n\to\infty} a_n = L\) and \(\lim_{n\to\infty} b_n = M\) then \[ \lim_{n\to\infty} \left( a_n \cdot b_n\right) = L\cdot M\text. \]
A special case of this is the following:
Constant factor rule If \(\lim_{n\to\infty} a_n = L\) and \(c\) is a constant then \[ \lim_{n\to\infty} c \cdot a_n = c\cdot L \]
For fractions, you can use the quotient rule:
Quotient rule If \(\lim_{n\to\infty} a_n = L\) and \(\lim_{n\to\infty} b_n = M\) with \(M\neq 0\) then \[ \lim_{n\to\infty} \frac{a_n}{b_n} = \frac{L}{M}\text. \]
If you want to calculate the limit of a fraction where the denominator tends to zero as \(n\to\infty\) you can try manipulating the fraction before applying the limit laws.
The proof of the multiplication law and quotient law are more technical so we will not prove them here.
Below you can find a short summary of the discussed calculation rules. Note that the details are missing: for example, you have to remember that in order to apply the quotient rule the denominator cannot converge to \(0\).
\[ \begin{aligned} \lim_{n\to\infty}\left( a_n + b_n\right) &= \lim_{n\to\infty} a_n + \lim_{n\to\infty} b_n \\[0.25cm] \lim_{n\to\infty}\left( a_n \cdot b_n\right) &= \lim_{n\to\infty} a_n \cdot \lim_{n\to\infty} b_n\\[0.25cm] \lim_{n\to\infty}c\cdot a_n &=c\cdot \lim_{n\to\infty} a_n \\[0.25cm] \lim_{n\to\infty} \frac{a_n}{b_n} &= \frac{\displaystyle\lim_{n\to\infty} a_n}{\displaystyle\lim_{n\to\infty} b_n}\text{ if }\lim_{n\to\infty}b_n\neq 0\end{aligned}\]
