1. L'Hopital's Rule

How would I use L'Hopital's Rule to calculate the following limits. I know the rule and can apply it, it's just that, I'm having trouble with some of these questions.

$1)\ \lim_ {x \rightarrow 0^+}\ x^\frac{2}{\ln x}$

$2)\ \lim_{x \rightarrow \infty}\ x^ \frac{1}{x}$

$2)\ \lim_{x \rightarrow \infty}\ a^ \frac{1}{x},\ a>0$

2. Originally Posted by acevipa
How would I use L'Hopital's Rule to calculate the following limits. I know the rule and can apply it, it's just that, I'm having trouble with some of these questions.

$1)\ \lim_ {x \rightarrow 0^+}\ x^\frac{2}{\ln x}$

$2)\ \lim_{x \rightarrow \infty}\ x^ \frac{1}{x}$

$2)\ \lim_{x \rightarrow \infty}\ a^ \frac{1}{x},\ a>0$
For these you will need to use the exponential-logarithmic transformation.

1) $x^{\frac{2}{\ln{x}}}= e^{\ln{\left(x^{\frac{2}{\ln{x}}}\right)}}$

$= e^{\left(\frac{2}{\ln{x}}\right)\ln{x}}$

$= e^2$.

Therefore $\lim_{x \to 0^{+}}x^{\frac{2}{\ln{x}}} = \lim_{x \to 0^{+}}e^2$

$= e^2$.

2) $x^{\frac{1}{x}} = e^{\ln{\left(x^{\frac{1}{x}}\right)}}$

$= e^{\left(\frac{1}{x}\right)\ln{x}}$

$= e^{\frac{\ln{x}}{x}}$.

Therefore $\lim_{x \to \infty}x^{\frac{1}{x}} = \lim_{x \to \infty}e^{\frac{\ln{x}}{x}}$

$= e^{\lim_{x \to \infty}\frac{\ln{x}}{x}}$

$= e^{\lim_{x \to \infty} \frac{\left(\frac{1}{x}\right)}{1}}$ by L'Hospital's Rule

$= e^{\lim_{x \to \infty}\frac{1}{x}}$

$= e^0$

$= 1$.

3) Surely $\lim_{x \to \infty}a^{\frac{1}{x}} = a^0 = 1$...