Monday, December 2, 2019

limits - Evaluating the limxto0frac4x18x1 without L'Hopital Rule




How to evaluate lim 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.


Answer



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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