Evaluating the Limit of f(x)/g(x) when limx→a g(x) = 0: Indeterminate and Non-Indeterminate Cases

If you’re trying to evaluate limx→a f(x)/g(x), and you first find that limx→a g(x) = 0, what else do you need to look at?

If you find that limx→a g(x) = 0, there are two possible cases that you need to consider in order to evaluate limx→a f(x)/g(x):

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If you find that limx→a g(x) = 0, there are two possible cases that you need to consider in order to evaluate limx→a f(x)/g(x):

1. Case 1: Indeterminate form of “0/0”:
If the limit of the numerator f(x) also approaches 0, i.e., limx→a f(x) = 0, then you have an indeterminate form of “0/0”. In this case, you need to further investigate the functions f(x) and g(x) to determine their behavior and find a way to simplify the expression f(x)/g(x) to evaluate the limit.

One approach to simplify the expression is to use algebraic manipulation or limit theorems such as L’Hôpital’s rule, which allows you to differentiate the numerator and denominator individually until you reach a form where the limit can be evaluated directly. L’Hôpital’s rule states that if the limit of f(x)/g(x) is in the form of “0/0” or “∞/∞”, then the limit of the derivative of f(x)/g(x) is equivalent to the original limit. By applying this rule iteratively, you can simplify the expression and evaluate the limit.

2. Case 2: Non-indeterminate form of “0/0”:
If the limit of g(x) is 0, but the limit of f(x) approaches a non-zero value, i.e., limx→a f(x) ≠ 0, then you have a non-indeterminate form of “0/0”. In this case, the limit may not exist. Differently put, if the denominator g(x) approaches zero while the numerator f(x) does not, the quotient f(x)/g(x) will approach infinity or negative infinity, depending on the behavior of f(x) as x approaches a.

To summarize, if you find that limx→a g(x) = 0, you need to carefully examine the behaviors of both f(x) and g(x) in order to determine the appropriate approach to evaluate the limit of f(x)/g(x).

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