An intro to assessing limits
The theory of the limit of a function goes back to Greek academicians like Eudoxus and Archimedes. While there were never any formal definitions of limits, a majority of their calculations were on the basis of this concept. Isaac Newton gave formal definition to the notion of a limit and Cauchy refined this concept. Limits make up the foundation of calculus, which then defines the foundation of several machine learning algorithms. Therefore, it is critical to understand how limits of differing types of functions are assessed.
In this blog post which serves as a functional tutorial, you will find out how to assess the limits of differing variants of functions.
Upon finishing this tutorial, you will be aware of:
- The differing rules for evaluating limits
- How to assess the limit of polynomials and rational functions
- How to assess the limit of a function with discontinuities
- The Sandwich Theorem
This tutorial is divided into three portions, they are:
- Rules for limits
- Instances of assessing limits leveraging the rules for limits
- Limits for polynomials
- Limits for rational expressions
- Limits for functions with a discontinuity
- The Sandwich Theorem
Rules for Limits
Limits are simple to assess if we are aware of a few simple principles, which are detailed below. All these rules have their basis on known limits of two functions f(x) and g(x), when x approaches a point k:
Instances of leveraging rules to assess limits
Here are a few instances that leverage the fundamental rules to assess a limit. Observe that these rules are applicable to functions which are defined at a point as x approaches that particular point.
Limits for polynomials
Instances 1 and 2 are that of polynomials. From the rules for limits, we can observe that for any polynomial, the limit of the polynomial when x gets close to a point k is equivalent to the value of the polynomial at k. It can be written as:
Therefore, we can assess the limit of a polynomial through direct substitution, for e.g.
Lim(xà1)x^4+3x^3+2 = 1^4+3(1)^3+2 = 6
Limits for rational function
For rational functions that consist of fractions, there are two scenarios. One case is assessing the limit when x approaches a point and the function has its definition at that point. The other scenario involves computing the limit when x approaches a point and the function is not defined at that point.
Scenario 1: Function is defined
Much like the scenario with polynomials, whenever we possess a function, which is a rational expression of the form f(x)/g(x) and the denominator is non-zero at a juncture, then:
Lim(xàk) f(x)/g(k) if g(k) is not equal to 0.
We can thus assess this limit through direct substitution. For instance:
Lim(xà0)(x^2+1)/(x-1) = -1
Here, we can apply the quotient rule or simpler still, substitute x=0 to assess the limit. However, this function has no restriction when x gets close to 1. Observe the first graph in the figure below.
Case 2: Function is not defined
Let’s observe another example:
At x=2 we are encountering an issue. The denominator equals zero, and therefore the function is not defined at x=2. We can observe from the figure that the graph of this function and (x+2) is the same, only not at the point x+2, where there is a hole. In this scenario, we can cancel out the mutual factors and still assess the limit for (xà2) as:
Lim(xà2)(x^2-4)/(x-2) = lim(xà2)(x-2)/(x-2) = lim(xà2)(x+2) = 4
The subsequent image illustrates the above two instances as well as a third similar instance of g_3(x):
Scenario for functions with a discontinuity
Assuming we have a function h(x), which has definitions for all real numbers:
h(x) = (x^2+x) / x if x is not equal to 0.
h(x) = 0, if x=0.
The function g(x) possesses a discontinuity at x equal to zero, as demonstrated in the subsequent figure. When assessing lim(xà0)h(x), we have to observe what occurs to h(x) when x is near 0 (and not when x = 0) As we get close to x=0 from either side, h(x) approaches 1, and therefore, lim(xà0)h(x)=1.
The function m(x) illustrated in the figure below is another fascinating case. This function also has definitions for all real numbers but the limit does not exist when xà0.
The Sandwich Theorem
This theorem is also referred to as the squeeze theorem or the pinching theorem. It says that when the following are true:
- x is close to k
- f(x) <=g(x) <=h(x)
- lim(xàk)f(x) = lim(xàk)h(x) = L
then the limit of g(x) as x approaches k is provided by:
lim (xàk)g(x) = L
This theorem is depicted in the figure that follows. Leveraging this theorem we can assess the limits of several complicated functions. A well known instance consists of the sine function: lim(xà0)x^2sin(1/x)
We are aware that the sin(x) always alternates between -1 and +1. Leveraging this fact, we can solve the limit as depicted below.
This section lists some concepts for extension of the tutorial that you may wish to look into:
- L’Hospitals Rule and Indeterminate Forms (needs function derivatives)
- Function derivative defined in terms of the limit of a function
- Function integrals