Indeterminate Forms: Overview, Questions, Preparation

"Indeterminate" means the unknown value. It is a mathematical expression, which is indeterminable even after the substitution of the limits. It is one of the crucial concepts of Limits and Derivatives. Here, Grade 12 students will learn that derivation of limits for some of the calculus operations can be in an indeterminate form. Let us have a look at the concept and examples. Sometimes, while solving mathematical problems, we cannot find out the solution for some arithmetic expressions. And, these expressions are known as indeterminate forms. It usually includes two fractions whose limits cannot be defined by referring to the individual functions' original limits.

These forms are prevalent in Calculus. 

Some of the indeterminate forms with their transformation and conditions are tabulated below:

What are the various methods to determine the results of indeterminate forms?

L's Hospital's Rule

This rule applies to 0/0 or ∞/∞ forms. As per this rule, it is an event of an indeterminate form, in which the numerator and denominator have to be differentiated separately for solving the problem.

The denominator and numerator derivatives are calculated individually after considering each step to check whether it is free from a variable or not, followed by making at least one of the terms constant.

Read more about: L's Hospital's Rule

Division of All the Terms by Highest Power

This method is usually applied on the indeterminate form of ∞/∞. Under this method, you have to divide both the denominator and numerator of the given problem by the variable’s highest power in the sum. After performing this method, we can easily calculate the limit value.

Factoring Method

It is the simplest form of calculating the limit of the indeterminate form of 0/0 form. In this method, the given problem is factorised into the maximum simplest factors. After deriving the simplest form, the limit value should be substituted. 

1. Simplify lim _x→∞ px2 + x − x.

Solution:

This is a ∞-∞ type of indeterminate form. Our first step is to remove the square root.

=√x²+x – x=(√x²+x–x) × √x²+x +x/√x²+x – x

=( x²+x) - x²/ √x²+x +x

= x/√x²+x +x

Then,

        =    Lim _x→∞  √x²+x – x

        =   Lim_x→∞x/√x²+x – x

        =      Lim_x→∞ 1/√1+x^-1 +1 

        =     1/2

Q: Enlist various types of indeterminate forms.

Q: Name the various methods for evaluating indeterminate forms?

Q: Can Hospital's methods apply to the terms having finite non-zero limits?

Q: Name the evaluation method that can be used for the 0/0 form?