An intro to indeterminate forms and L’Hospital’s Rule
Indeterminate forms are typically encountered when assessing limits of functions, and limits in turn have a critical part within calculus and mathematics. They are critical for learning all about gradients, derivatives, Hessians, and a ton more.
In this guide by AICoreSpot, you will find out how to assess the limits of indeterminate forms and the L’Hospital’s rule for finding solutions to them.
After finishing this tutorial, you will be aware of:
- How to assess the limits of functions possessing indeterminate types of the form 0/0 and infinite/infinite
- L’Hospitals Rule for assessing indeterminate types
- How to translate more complicated indeterminate types and apply L’Hospital’s rule to them
This guide is subdivided into two portions, they are:
- The indeterminate forms of type 0/0 and infinity/infinity
- How to apply L’Hospital’s rule to these types
- Solved instances of these two indeterminate types
- More complicated indeterminate types
- How to translate the more complicated indeterminate types to 0/0 and infinity/infinity forms
- Solved instances of such types
This guide necessitates a fundamental understanding of the following two subjects:
- Limits and Continuity
- Assessing limits
Don’t worry, we’ll be covering these in future posts on AICoreSpot.
What are indeterminate forms?
When assessing limits, we come across scenarios where the fundamental rules for assessing limits might fail. For instance, we can go about applying the quotient rule with regards to rational functions.
lim(x→a) f(x)/g(x) = (lim(x→a)f(x))/(lim(x→a)g(x)) if lim(x→a)g(x)≠0
The above specified rule can only have application if the expression in the denominator does not approach zero as x get close to a. A more complex scenario props up if both the numerator and denominator both approach zero as x approaches a. This is referred to as an indeterminate form of type 0/0. Likewise, there are indeterminate forms of the type ∞/∞ provided by:
lim(x→a) f(x)/g(x) = (lim(x→a)f(x))/(lim(x→a)g(x)) when lim(x→a)f(x)=∞ and lim(x→a)g(x)=∞
What is L’Hospitals Rule?
The L’Hospital Rule specifies the following:
Application of L’Hospital’s Rule
A critical point to observe is that L’Hospital’s rule is only relevant when the conditions for f(x) and g(x) are met. For instance:
- lim(𝑥→0) sin(x)/(x+1) Cannot apply L’Hospital’s rule as it’s not 0/0 form
- lim(𝑥→0) sin(x)/x Can apply the rule as it’s 0/0 form
- lim(𝑥→∞) (e^x)/(1/x+1) Cannot apply L’Hospital’s rule as it’s not ∞/∞ form
- lim(𝑥→∞) (e^x)/x Can apply L’Hospital’s rule as it is ∞/∞ form
Instances of 0/0 and ∞/∞
Some instances of these two variants, and how to find solutions to them are displayed underneath. You can also go about referring to the figure below to refer to these functions.
Example 1.1: 0/0
Assess lim(𝑥→2) ln(x-1)/(x-2) (Observe the left graph in the figure)
Example 1.2 : ∞/∞
Evaluate lim(𝑥→∞) ln(x)/x (See the right graph in the figure)
Even more indeterminate forms
The L’Hospital rule informs us how to handle 0/0 or ∞/∞ forms. But, there are additional indeterminate forms that consist of products, differences, and powers. So, how do we tackle the remainder? We can leverage some smart tricks within mathematics to convert products, differences, and powers into quotients. This can facilitate us to easily apply L’Hospital Rule to nearly all indeterminate forms. The table underneath displays several intermediate forms and how to handle them.
The following instances display how you can convert one indeterminate for to either 0/0 or ∞/∞ form and go about applying L’Hopsital’s rule to solve the limit. After the worked out instances, you can also observe the graphs of all the functions whose limits have been calculated.
Example 2.1: 0.∞
Evaluate lim(𝑥→∞) x.sin(1/x) (See the first graph in the figure)
Example 2.2: ∞-∞
Evaluate lim(𝑥→0) 1/(1-cos(x)) – 1/x (See the second graph in the figure below)
Example 2.3: Power Form
Evaluate lim(𝑥→∞) (1+x)^(1/x) (See the third graph in the figure below)
This section details a few ideas and concepts for extending the guide that you may desire to explore.
- Cauchy’s Mean Value Theorem
- Rolle’s Theorem