<s>[INST] Find $\lim_{x\rightarrow0}\frac{3^x-5^x}{4^x-10^x}$ Find $$\lim_{x\rightarrow0}\frac{3^x-5^x}{4^x-10^x}$$
My work so far:
$$\lim_{x\rightarrow0}\frac{3^x-5^x}{4^x-10^x}=\frac{\ln3-\ln5}{\ln4-\ln10}$$
Is correct?
Add:
I used $a^x\sim 1+x\ln a$ for $x\rightarrow 0$
 [/INST] The expression equals
$$\frac{(3^x-3^0) - (5^x-5^0)}{(4^x - 4^0)-(10^x-10^0)}.$$
Divide top and bottom by $x=x-0.$ Then by definition of the derivative, the desired limit equals
$$ \frac{(3^x)'(0) - (5^x)'(0)}{(4^x)'(0)- (10^x)'(0)}= \frac{\ln 3 - \ln 5}{\ln 4 - \ln 10}.$$
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