Question

Assertion $(\mathbf{A}): \lim _{\mathrm{x} \rightarrow 0^{\circ}}\left(\mathrm{x}^{\mathrm{x}^{*}}-\mathrm{x}^{\mathrm{x}}\right)=-1$ Reason \((\mathbf{R}): \lim _{x \rightarrow 0^{\prime}} x^{x}(x-1)=-1\)

Short answer

#tag_title# Short answer based on step-by-step solution: #tag_content# The reason provided, \(\lim_{x \rightarrow 0^+} (x^x)(x-1) = -1\), is not valid because the natural logarithm of the limit in step 4 results in an undefined value (\(\ln(-1)\)). Consequently, this reason does not directly support the assertion. The correct limit needs to be found using other techniques.

Step-by-step solution

Rewrite the assertion in terms of limits.

Since the assertion is given in limit notation, rewrite it like this: \(\lim _{x \rightarrow 0^+}\left(x^{x^*} - x^x\right) = -1\).

Rewrite the reason in terms of limits.

Rewrite the reason as a limit statement: \(\lim _{x \rightarrow 0^+} (x^x)(x-1) = -1\).

Check the indeterminate form.

In order to check the indeterminate form, let's analyze the functions \(x^{x^*}\) and \(x^x\). As x approaches \(0\) from the positive side, both base (x) and exponent (either \(x^*\) or x) approaches 0. The form \(0^0\) is an indeterminate form.

Apply L'Hôpital's rule.

Since the indeterminate form is \(0^0\) in this case, we cannot directly apply L'Hôpital's rule. However, we can use logarithmic properties and L'Hôpital's rule to analyze the behavior of the limit. For this, we take the natural logarithm of the limit: \(\ln\left(\lim _{x \rightarrow 0^+} (x^x)(x-1)\right) = \ln(-1)\) However, the natural logarithm of \(-1\) is undefined, so the reason provided is not valid.

Analyze the correct form

To analyzethe correct form of the expression, we need to find the limit of the given expression. Since their individual limits are indeterminate, we will focus on the difference \(x^{x^*} - x^x\). Now, let's replace \(x^*\) by its definition: \(x^* = \lim_{n \rightarrow \infty} \frac{x^n}{n}\). Thus, \(\lim _{x \rightarrow 0^+} \left(x^{\lim_{n \rightarrow \infty} \frac{x^n}{n}} - x^x\right)\) Now, use properties of limits to find the correct limit: \(\lim_{n \rightarrow \infty} \lim _{x \rightarrow 0^+} \left(x^{\frac{x^n}{n}} - x^x\right)\)

Find the correct limit

The correct limit is yet to be found, but from our analysis, we can see that the reason provided in the exercise does not lead us to the assertion directly. Further analysis should involve exploring other techniques to find the correct limit.