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.
Wt is 0 (x) can anyone ans
that is e(x)
what is o(x)