Stewart makes some L'Hospital simplifcations that do not totally make sense to me. Here are 3:
1.
How does Stewart get form the highlighted second step the third step? The book isn't very clear...
Same with this one. The derivation isn't very clear. Can someone flesh it out a bit for me?
3.
Also this one:
How do we get to:
$$\lim_{x \to \inf} \frac{\frac{1}{x}}{\frac{1}{2\sqrt{x}}} $$
On
To go from the yellow highlighted step to the next step in these problems, apply L'Hospital's Rule -- take derivatives of top and bottom separately...
On
L'Hospital's rule applies to a limit if it is in the following form: $\lim_{x \rightarrow c} \frac{f}{g}$, when either both $\lim_{x \rightarrow c} f = \pm \infty$ and $\lim_{x \rightarrow c} g = \pm \infty$, or both $\lim_{x \rightarrow c} f = 0$ and $\lim_{x \rightarrow c} g = 0$.
As you can see in the first example, $\lim_{x \rightarrow 0}xlnx = 0 \cdot - \infty$, and L'Hospital's rule does not apply. So, we change $x$ to the equivalent $\frac{1}{\frac{1}{x}}$. Now the limit of the function is $\frac{- \infty}{\infty}$ making it possible to apply L'Hospital's rule.
In the first highlighted part, you can say that x=1/(1/x) because dividing by a fraction means multiplying by the reciprocal of that fraction, so 1/(1/x)=1*x/1=x
In the second problem, he shows most of the steps. First, he does a cross multiplication to make the two fractions into one. Then, he goes straight into a derivation of the top and bottom components of the fraction. After that, he multiplies the fraction by x/x to clean things up (it will be an equivalent fraction, since x/x=1). Finally, he does L'Hospital's rule again and simplifies.
In the third problem, he first does L'Hospital's rule. He then does something similar to problem 1, where he makes the fraction tidier by moving things where they belong (since the numerator is a fraction, and the denominator is a fraction, all he has to do is multiply the numerator by the reciprocal of the denominator to get 2/sqrt(x).) And since the infinite limit of 2 is 2, you can't continue with L'Hospital's rule.