Question

Explain with examples what is meant by the indeterminate form \(0 / 0\)

Short answer

Answer: An indeterminate form is an undefined or ambiguous expression that occurs when trying to evaluate the limit of a function, such as the \(0/0\) form. To resolve indeterminate forms, one can use methods like simplification or algebraic techniques to cancel out the problematic terms that are tending to zero, allowing proper evaluation of the limit.

Step-by-step solution

Introduction to Indeterminate Forms

Indeterminate forms occur when the expression we are trying to evaluate has a form that is undefined or ambiguous. In this case, we are focusing on the indeterminate form \(0/0\). This occurs when both the numerator and denominator of a fraction tend to zero as the limit approaches a particular point.

Example 1: \(0 / 0\) Indeterminate Form

Consider the limit: $$ \lim_{x \to 0} \frac{x^2}{x} $$ To evaluate this limit directly, we would plug in \(x=0\) into the expression: $$ \frac{0^2}{0} = \frac{0}{0} $$ Since we cannot divide by zero and the expression \(0/0\) is undefined, we have encountered an indeterminate form, \(0/0\).

Example 2: Another \(0 / 0\) Indeterminate Form

Another example of an indeterminate form is the limit: $$ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} $$ Trying to evaluate this limit directly, we get: $$ \frac{1^2 - 1}{1 - 1} = \frac{0}{0} $$ Once again, we are faced with the indeterminate form \(0/0\).

Resolving Indeterminate Forms Using Simplification

To properly resolve the indeterminate forms \(0/0\), we often use methods that simplify the expressions before evaluating the limit. Using algebra, we can often cancel out the problematic terms that are tending to zero. For example 1, we can simplify the expression: $$ \frac{x^2}{x} = x $$ Now, the limit can be evaluated directly: $$ \lim_{x \to 0} x = 0 $$ For example 2, we can factor the numerator and cancel the common terms: $$ \frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1 $$ Now, the limit can be evaluated directly: $$ \lim_{x \to 1} (x + 1) = 2 $$ In both cases, simplifying the expressions allowed us to resolve the indeterminate forms and find the proper limit of the functions.