How To Solve 0 ⋅ ∞ Indeterminate Forms In Calculus: Techniques And Examples

to find the limit of a 0 ⋅ ∞ indeterminate form…

rewrite expression as 0/0 or ∞/∞, then apply L’Hopital’s rule

To find the limit of a 0 ⋅ ∞ indeterminate form, you need to rewrite the original expression in a way that makes it easier to evaluate. There are a couple of common techniques for doing this:

1. Rewrite the expression using algebraic manipulation. For example, you can often multiply the expression by a factor that allows you to use properties of limits that you already know. For instance, if you have an expression of the form 0 ⋅ ∞, you could multiply the expression by 1/∞ and then take the limit to get 0. Similarly, if you have an expression of the form n ⋅ ∞, you could use algebraic manipulation to rewrite it as ∞/1 over n, and then take the limit to get ∞.

2. Apply L’Hopital’s rule. In certain cases, you can apply L’Hopital’s rule to find the limit of a 0 ⋅ ∞ indeterminate form. This rule says that if you have a limit of the form f(x)/g(x) where f(x) and g(x) both approach 0 or ∞ as x approaches some value, and if the limit of the ratio of their derivatives f'(x)/g'(x) exists, then the limit of the original function also exists and is equal to the limit of the ratio of their derivatives. You can apply L’Hopital’s rule by taking the derivative of both the numerator and denominator of the original function with respect to x until you no longer have an indeterminate form.

For example, let’s say we want to find the limit of f(x) = x ⋅ ln(x) as x approaches 0+. The limit of this function is a 0 ⋅ ∞ indeterminate form. To solve it, we can use L’Hopital’s rule as follows:

f(x) = x ⋅ ln(x)

f'(x) = ln(x) + 1

f”(x) = 1/x

lim x->0+ f(x) = lim x->0+ [ln(x) + 1] / (1/x)

Now, we have an expression of the form ∞/∞, which is still indeterminate. However, we can apply L’Hopital’s rule again:

lim x->0+ [ln(x) + 1] / (1/x) = lim x->0+ 1/x / (-1/x^2) = lim x->0+ -x = 0

Therefore, the limit of the original function is 0.

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