Calculation rules for limits

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:

If $\lim_{n\to\infty} a_n = L$ and $\lim_{n\to\infty} b_n = M$ then we have

$\lim_{n\to\infty} \left( a_n + b_n\right) = L+M\text.$

Assume that $\lim_{n\to\infty} a_n = L$ and $\lim_{n\to\infty} b_n = M$ and let $\epsilon > 0$ be given. According to the definition of $\lim_{n\to\infty} a_n = L$ there exists an $N_1$ so that for all $n>N_1$ it holds that $|a_n - L|<\frac{\epsilon}{2}$ . Similarly, there exists $N_2$ such that for all $n>N_2$ we have $|b_n - M|<\frac{\epsilon}{2}$ . The triangle inequality now implies that for all $n > \max\{N_1, N_2\}$ the inequality

$\left|(a_n + b_n) - (L+M)\right| \leq \left| a_n - L \right| + \left| b_n - M \right| < \frac{\epsilon}{2}+ \frac{\epsilon}{2}=\epsilon$ holds. This proves that

$\lim_{n\to\infty} \left( a_n + b_n\right) = L+M\text.$

There is also a 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:

If $\lim_{n\to\infty} a_n = L$ and $c$ is a constant then

$\lim_{n\to\infty} c \cdot a_n = c\cdot L$

Let $\epsilon > 0$ be given, then there exists an $N$ such that for all $n > N$ we have $|a_n - L|<\frac{\epsilon}{c}$ . If we multiply the left- and right-hand side by $c$ we get the desired result.

For fractions, you can use the 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 $0$ .

$\begin{aligned} \lim_{n\to\infty}\left( a_n + b_n\right) &= \lim_{n\to\infty} a_n + \lim_{n\to\infty} b_n \\[0.25cm] \lim_{n\to\infty}\left( a_n \cdot b_n\right) &= \lim_{n\to\infty} a_n \cdot \lim_{n\to\infty} b_n\\[0.25cm] \lim_{n\to\infty}c\cdot a_n &=c\cdot \lim_{n\to\infty} a_n \\[0.25cm] \lim_{n\to\infty} \frac{a_n}{b_n} &= \frac{\displaystyle\lim_{n\to\infty} a_n}{\displaystyle\lim_{n\to\infty} b_n}\text{ if }\lim_{n\to\infty}b_n\neq 0\end{aligned}$