Limit Basics

Today's tutoring session with Chris was mostly about series and sequences, but a key concept in understanding them is the idea of a limit. Limits are also fundamental to differential and integral calculus. What is a limit? A limit describes the behavior of a sequence or a function as the variable approaches some number, usually infinity or zero. Lets look at some examples:

1) What is the limit of the sequence {1, 1/2, 1/3, 1/4,...,1/n} as n approaches infinity?

We can see that as n gets larger the number gets smaller. So as n goes out towards infinity the number becomes infinitesimally smaller coming closer and closer to zero, and so we say that the limit is zero. This sequence converges to zero.

2) What is the limit of the sequence {1, 4, 9, 16, 25,...,n^2} as n approaches infinity?

We can see that as n gets larger so does the number--exponentially. So as n goes out towards infinity the number becomes infinitely larger, and so we say that the limit is infinity. The limit diverges.

3) What is the limit of the sequence {cos(π), cos(2π), cos(3π),...,cos(nπ)} as n approaches infinity?

From trigonometry we know that this is equivalent to {1, -1, 1, -1,...}, and so this sequence oscillates forever between 1 and -1. This limit does not exist and is considered divergent.

To denote a limit symbolically we write lim n→∞ which is read as "the limit as n approaches infinity." (It is important to note that on paper "n→∞" is written underneath "lim" however we cannot do that on a computer).

These are simple examples. What if the sequence is explicitly defined by a rational function? There are three simple rules that will allow such a limit to be found quickly. Lets take a rational function defined by (a^n+c)/(b^m+d)

*If n=m, then the limit is the ratio of the leading coefficients.
*If n>m, then the limit diverges to infinity
*If n<m, then the limit converges to zero.

 For example:

1) lim n→∞ (3n+1)/(n-2)  = 3/1 = 3       
2) lim n→∞ (n^3)/(n^2+10)  = infinity and is divergent      
3) lim n→∞ (3-n)/(n^2+5) = 0

Limits are fundamental to calculus. Both derivatives (rates of change) and integrals (areas under curves) are driven by limits. This shall be discussed further in another post.

-mike-