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  • Evaluate  <img alt="\mathrm{\lim _{x \rightarrow 1^{+}} \frac{x^{x^x}-x^x}{x-1}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bx%20%5Crightarrow%201%5E%7B

Evaluate  \mathrm{\lim _{x \rightarrow 1^{+}} \frac{x^{x^x}-x^x}{x-1}}

Option: 1

0


Option: 2

3


Option: 3

5


Option: 4

1


Answers (1)

best_answer

Taking x^{x} as a factor in numerator we can see that the limit in question is equal to 

\mathrm{\lim _{x \rightarrow 1^{+}} \frac{x^{x^x-x}-1}{x-1}}

and this can be further expressed as 

\mathrm{\lim _{x \rightarrow 1^{+}} \frac{\exp \left(\left(x^x-x\right) \log x\right)-1}{\left(x^x-x\right) \log x} \cdot\left(x^x-x\right) \cdot \frac{\log x}{x-1}}

and the desired limit is clearly equal to 1.0.1=0

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mansi

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