Understanding Indeterminate Forms in Mathematics | Exploring Limit Evaluations

What does it mean for a limit to be indeterminate? Give two examples of indeterminate expressions.

In mathematics, a limit is considered indeterminate when it does not have a clear or definite value

In mathematics, a limit is considered indeterminate when it does not have a clear or definite value. This occurs when evaluating certain expressions where the limit cannot be determined solely based on the given information. The indeterminate form arises in situations where different factors or variables of the expression approach different values simultaneously, making it impossible to determine the outcome just by looking at the expression itself.

Here are two examples of indeterminate expressions:

1. The expression 0/0: When evaluating the limit of a function as it approaches a certain value, if both the numerator and denominator approach zero, it becomes challenging to determine the value of the expression. For example, consider the limit of (x^2 – 4)/(x – 2) as x approaches 2. Plugging in x = 2 directly would result in the indeterminate form 0/0.

2. The expression ∞ – ∞: In some cases, when dealing with infinites, the result may be indeterminate. For instance, consider the limit of (x^2 + x) – (x^2 – x) as x goes to infinity. Both terms individually tend to infinity, but the result is undefined since there isn’t a clear numerical difference between the two infinities, resulting in an indeterminate form.

When encountering indeterminate forms, further mathematical techniques such as L’Hôpital’s rule, factoring, or simplification may be used to transform the expression into a determinate form that allows for evaluation.

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