Indeterminate Limit: Evaluating ##\displaystyle \lim_{a \to 0^+} a^2 \log a##

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In summary, an indeterminate limit is a limit that cannot be immediately evaluated by substitution or basic algebraic manipulation. To evaluate an indeterminate limit, more advanced techniques such as L'Hopital's rule, Taylor series, or other limit laws are typically used. When the limit is approaching 0 from the right (a→0+), it means that the input (a) is getting closer to 0 from the positive side of the number line. The expression a^2 log a is a common example of an indeterminate limit and has no defined value at a=0. To evaluate this limit, one can rewrite the expression and use L'Hopital's rule, resulting in a limit of 0.
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Mr Davis 97
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##\displaystyle \lim_{a \to 0^+} a^2 \log a = 0 \cdot (- \infty)##, which is an indeterminate form.

So ##\displaystyle \lim_{a \to 0^+} a^2 \log a = \lim_{a \to 0^+} \frac{\log a}{a^{-2}} = \lim_{a \to 0^+} \frac{\frac{1}{a}}{(-2)a^{-3}} = -\frac{1}{2}\lim_{a \to 0^+} a^2 = 0##.

Is this correct?
 
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Yes. I guess you are learning about Hospital's rule.
 
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1. What is an indeterminate limit?

An indeterminate limit is a limit that cannot be immediately evaluated by substitution or basic algebraic manipulation. In other words, the value of the limit is not obvious from just looking at the expression.

2. How do you evaluate an indeterminate limit?

To evaluate an indeterminate limit, you typically need to use more advanced techniques such as L'Hopital's rule, Taylor series, or other limit laws. These methods involve taking derivatives or approximating the function to find the limit value.

3. What does it mean when the limit is approaching 0 from the right (a→0+)?

This means that the value of the limit is being evaluated as the input (a) approaches 0 from the positive side of the number line. In other words, a is getting closer and closer to 0 without actually reaching it.

4. What is the significance of the expression ##a^2 \log a## in the indeterminate limit ##\displaystyle \lim_{a \to 0^+} a^2 \log a##?

The expression ##a^2 \log a## is a common example of an indeterminate limit. It is often used in calculus courses to demonstrate the need for more advanced techniques in evaluating limits. The expression itself has no defined value at a=0, which is why the limit is indeterminate.

5. Can you provide an example of how to evaluate the indeterminate limit ##\displaystyle \lim_{a \to 0^+} a^2 \log a##?

One way to evaluate this limit is to rewrite the expression as ##\displaystyle \lim_{a \to 0^+} \frac{\log a}{\frac{1}{a^2}}## and then use L'Hopital's rule to find the limit. This results in a limit of 0, indicating that the function ##a^2 \log a## approaches 0 as a approaches 0 from the right.

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