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I don't see the way to solve this limit. $$\lim_{x \to 0}\frac{3^{5x}-2^{7x}}{\arcsin\left(2x\right) - x} $$ My attempt is 1) Divide the numerator by $3^{5x}$

$$\lim_{x \to 0}\frac{3^{5x}-2^{7x}}{\arcsin\left(2x\right) - x} = \lim_{x \to 0} \frac {1- \left(2/3 \right)^{5x}\,\,2^{2x}}{\arcsin\left(2x\right)-x} $$ And I am unable to do next step. So how this limit can be solved?

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Use the fact that

$$3^{5 x}-2^{7 x} = e^{(5 \log{3}) x} - e^{(7 \log{2}) x}$$

and that, for small $y$

$$e^{y} = 1 + y + \cdots$$

and further, for small $z$

$$\arcsin{z} = z + \cdots$$

so that, for $x$ near $0$:

$$\frac{3^{5 x}-2^{7 x}}{\arcsin{(2 x)}-x} \sim \frac{(5 \log{3}-7 \log{2}) x}{2 x-x} = 5 \log{3}-7 \log{2}$$

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You can write $$\frac{3^{5x}-2^{7x}}{\arcsin 2x-x}=\frac{3^{5x}-1}{\arcsin 2x-x}-\frac{2^{7x}-1}{\arcsin 2x-x}=$$$$\left(5\frac{3^{5x}-1}{5x}-7\frac{2^{7x}-1}{7x}\right)\left(2\frac{\arcsin 2x}{2x}-1\right)^{-1}.$$ Note then that all the fractions are expressed as derivatives of the functions involved, as $x\to 0$.

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$\newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\dd}{{\rm d}}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\ic}{{\rm i}}% \newcommand{\imp}{\Longrightarrow}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert #1 \right\vert}% \newcommand{\yy}{\Longleftrightarrow}$ $\large \mbox{No H'opital, just Taylor}:$

$$ {3^{5x} - 2^{7x} \over \arcsin\pars{x} - x} = {% {\bracks{1 + \ln\pars{3^{5}}x} - \bracks{1 + \ln\pars{2^{7}}x}} \over \pars{2x} - \pars{x}} + \color{#ff0000}{\Large\cdots} = {5\ln\pars{3} - 7\ln\pars{2}} + \color{#ff0000}{\Large\cdots} $$

where "$\color{#ff0000}{\Large\cdots}$" terms go to zero cuando $x \to 0$.

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Using L'Hospital's rule, we find

\begin{align*} \lim_{x \to 0} \frac{3^{5x} - 2^{7x}}{\arcsin{2x} - x} &= \lim_{x \to 0} \frac{3^{5x} (5 \ln{3}) - 2^{7x}(7 \ln{2})}{\frac{2}{\sqrt{1 - (2x)^2}} - 1} \\ &= \frac{3^0 (5 \ln{3}) - 2^0 (7 \ln{2})}{2 - 1} \\ &= 5 \ln{3} - 7 \ln{2} \end{align*}

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  • $\begingroup$ thanks. but how this can be solved without L'Hospital's rule? $\endgroup$
    – k1ber
    Oct 25 '13 at 8:55

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