Evaluate the limit for 
.
![\[ \lim_{x \to 0} \frac{a^x - 1}{b^x - 1}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-5a92f81dae9cbb85475641e30d81074c_l3.png "Rendered by QuickLaTeX.com")
First we write 
and 
. Then we use the expansion of 
(p. 287), to obtain expansions for 
and 
,
![\[ a^x = e^{x \log a} = 1 + (x \log a) + o (x), \qquad b^x = e^{x \log b} = 1 + (x \log b) + o(x). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6505348419a4e9a853338123e649bf45_l3.png "Rendered by QuickLaTeX.com")
Therefore, we have
![\begin{align*} \lim_{x \to 0} \frac{a^x - 1}{b^x - 1} &= \lim_{x \to 0} \frac{1 + x \log a + o(x) - 1}{1 + x \log b + o(x) - 1} \\[9pt] &= \lim_{x \to 0} \frac{x \log a + o(x)}{x \log b + o(x)} \\[9pt] &= \lim_{x \to 0} \frac{\log a + \frac{o(x)}{x}}{\log b + \frac{o(x)}{x}} \\[9pt] &= \frac{\log a}{\log b}. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-70eff52af5a6884766967a899cd2b5e9_l3.png "Rendered by QuickLaTeX.com")
Evaluate the limit for .
First we write and . Then we use the expansion of (p. 287), to obtain expansions for and ,
Therefore, we have
For those asking, o(x) is little o notation and it means different things in different cases but here it means *some* function f(x) such the quotient f(x)/x approaches 0 as x approaches 0. The definition in Apostol’s book was stated briefly and then Apostol started abusing the notation without explaining everything.
For real!! This is my first exposure to the notation and there are so many things I have been confused about/had to sort out on my own.
Wt is 0 (x) can anyone ans
that is e(x)
what is o(x)