Limits are an operation on functions that are used frequently in engineering and scientific mathematics. For many well- behaved functions, this operation seems unnecessary, but to introduce this idea of limits, I will begin with a well-behaved function.
Consider the function f(x) = x2. What does f(x) approach as x gets approaches 1?
Well this seems to be pointless as the limit of this function as x approaches 1 is exactly what the function is equal to at 1. That is the limit is f(1) = 1. But the word “approach” used before is very important when it comes to limits as seen in the next example.
Consider the piecewise function (explained in my last post):
The graph of this function is
This function is exactly the same as the first one except that at x = 1, the function is equal to 2 and not the x2 value of 1. This is indicated by the open circle on the parabola. Now even though the value of the function has changed, the limit of this function as x approaches 1 is still 1, not 2 which is the value of the function. That is because limits are the value we approach as the variable gets closer and closer to a number without actually getting there.
Before I continue, I am tired of writing”what is the limit of a function as a variable in the function approach some value”. As usual, mathematicians have a shortcut:
This is read “the limit of f(x) as x approaches a“.
Now let’s look at
Now consider the limit of this function as x approaches some number a. For a equal to any number other than 1, the limit is just the value of the function f(a). But if a = 1, there’s a problem. If we approach 1 from the right, we get an answer of 2. If we approach from the left, we get an answer of 1. If this happens, the limit is said to not exist. By the way, there are left and right hand limits in mathematics, but these are beyond the scope of this post.
Now a can be infinity as well. Consider the hyperbola f(a) = 1/x:
One last example. You may be familiar with Euler’s number e which is irrational like the number š. It is used extensively in calculus and the technical areas that use calculus. However, this number has its origins in finance. When money is invested and compounded, that is the interest earned is added to the amount invested before the next round of interest is added, the more frequently that the interest is added during a year, the more you make in a year. Interest can be compounded yearly, quarterly, monthly, and as the interest is compounded more frequently, the more money you make. So Jacob Bernoulli asked the question “What is the maximum amount one can make as the frequency of compounding is increased?”. If you invest $1 at 100% annual interest which is compounded n times a year, at the end of 1 year you will have
dollars. So in mathematics terms, Bernoulli asked this question and it was Leonhard Euler who equated the answer as the number e:
So the maximum amount you will have after investing $1 at 100% interest compounded continuously for 1 year is $2.72.
Evaluating limits of complex functions can be difficult near troublesome values of a. I will look at some examples in my next post.