Indeterminate Forms | Engineering Mathematics for Mechanical Engineering PDF Download

What is Indeterminate Form?

Indeterminate Forms of Limits

$If\; we\; have\; the\; limits\; like,$

If the limits are applied for the given function, then it becomes 0/0, which is known as indeterminate forms.

In Mathematics, there are seven indeterminate forms that include 0, 1 and ∞, They are

0/0, 0×∞,∞/∞, ∞ −∞, ∞⁰, 0⁰, 1^∞.

Indeterminate Forms List

Indeterminate Form 1

Indeterminate Form 2

Indeterminate Form 3

Indeterminate Form 4

Indeterminate Form 5

Indeterminate Form 6

Indeterminate Form 7

How to Evaluate Indeterminate Forms?

1. Factoring Method (0/0 form): In the factoring method, the expressions are factored to their maximum simplest form. After that, the limit value should be substituted.
2. L’Hospital’s Rule (0/0 or ∞/∞ form): In this method, the derivative of each term is taken in each step successively until at least one of the terms becomes free of the variable. It means that at least one term becomes constant.
3. Division of Each Term by Highest Power of Variable (∞/∞ form): In this method, each term in numerator and denominator is divided by the variable of the highest power in the expression, and then, the limit value is obtained.
   

Solved Numericals

$Question:\; Evaluate$Q1. Evaluate

Solution: $Given:$Given,

Let f(x) = sin 2(x) and g(x) = e^x. + x
Therefore, f’(x) = 2 cos 2x , g’(x) = e^x + 1

$Therefore,$

Now, substitute the limits, it becomes

= 2 cos(0)/ e⁰ + 1

= 2/ 2 = 1

Therefore  = 1$herefore,$