Calculus: L′ Hopital's Rule

Question

1. $\displaystyle\lim_{x\to\frac{\pi}{2}}(\sin x)^{\tan x}$

2. $\displaystyle\lim_{x\to0}x^2\ln x$

3. $\displaystyle\lim_{x\to1^+}x^{\frac{1}{1-x}}$

Do I have to apply l'Hôpital's Rule to evaluate these limits?

Answer 1

In 1. and 3., take the logarithm of the function and then write it as a fraction.

In 2., $x^2\ln x=\dfrac{x^2}{\frac{1}{\ln x}}$.

Answer 2

For the second limit: Yes $$\lim_{x\to 0}x^2\log x=\lim_{x\to 0}\frac{\log x}{\frac1{x^2}}=\lim_{x\to 0}\frac{\frac 1x}{-2x^{-3}}=-\lim_{x\to 0}\frac 12x^2=0$$ and for the other limits it's more simple to use the Taylor series.