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How to evaluate $\displaystyle \lim_{x\to0}\left(\frac{4^x-1}{8^x-1}\right)$ without L'Hopital rule?

When I evaluated this limit I got an indetermination, $\frac{0}{0}$. I learned that in a rational function when one get $\frac{0}{0}$ indeterminated form, one should look for the common terms between numerator and denominator by factoring. But I can't figure out how to find the common terms in this case. Can you help me? Thanks.

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Note that $$4^x-1=(2^2)^x-1=(2^x)^2-1=\color{red}{(2^x-1)}(2^x+1)$$ $$8^x-1=(2^3)^x-1=(2^x)^3-1=\color{red}{(2^x-1)}(2^{2x}+2^x+1)$$

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Hint We can factor $a^{k x} - 1$ as: $$a^{k x} - 1 = (a^x - 1)(a^{(k - 1) x} + a^{(k - 2) x} + \cdots + a^x + 1).$$

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HINT:

As $\lim_{h\to0}\dfrac{e^h-1}h=1,$ and as $ a=e^{\ln(a)}$

$$\lim_{h\to0}\dfrac{a^h-1}h=\ln(a)\cdot\lim_{h\to0}\dfrac{e^{h\ln(a)}-1}{h\ln a}=\ln a$$

Now we know $\ln(b^m)=m\ln(b)$ when both logarithm remain defined

For real calculus we need $a>0$

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$$\displaystyle \lim_{x\to0}\frac{4^x-1}{8^x-1}=\lim_{x\to0} (\frac{4^x-1}{x}.\frac{x}{8^x-1})$$

now :

$$\lim_{x\to0}\frac{4^x-1}{x}=\ln 4$$

$$\lim_{x\to0}\frac{x}{8^x-1}=\lim_{x\to0}\frac{1}{\frac{8^x-1}{x}}=\frac{1}{\ln8}$$

so :

$$\lim_{x\to0}\frac{4^x-1}{8^x-1}=\frac{\ln 4}{\ln8}=\frac{2\ln 2}{3\ln 2}=\frac{2}{3}$$

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  • 1
    $\begingroup$ change 3ln in denominator to 3 ln 2 $\endgroup$ – user12345 Apr 29 '17 at 10:42

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