Answer
$\displaystyle\lim_{x\rightarrow 0} \dfrac{3^x-1}{x}=\ln 3$
$\displaystyle\lim_{x\rightarrow 0} \dfrac{5^x-1}{x}=\ln 5$
$\displaystyle\lim_{x\rightarrow 0} \dfrac{b^x-1}{x}=\ln b$
Work Step by Step
We have to determine:
$\displaystyle\lim_{x\rightarrow 0} \dfrac{b^x-1}{x}$
Compute the limit for $b=3$:
$\dfrac{3^{-0.1}-1}{-0.1}\approx 1.0404154$
$\dfrac{3^{-0.01}-1}{-0.01}\approx 1.0925995$
$\dfrac{3^{-0.001}-1}{-0.001}\approx 1.098009$
$\dfrac{3^{-0.0001}-1}{-0.0001}\approx 1.09985519$
$\dfrac{3^{-0.00001}-1}{-0.00001}\approx 1.0986063$
$\dfrac{3^{0.00001}-1}{0.00001}\approx 1.0986183$
$\dfrac{3^{0.0001}-1}{0.0001}\approx 1.0986726$
$\dfrac{3^{0.001}-1}{0.001}\approx 1.099216$
$\dfrac{3^{0.01}-1}{0.01}\approx 1.1046692$
$\dfrac{3^{0.1}-1}{0.1}\approx 1.1612317$
$\ln 3\approx 1.09861228867$
Therefore we got:
$\dfrac{3^{x}-1}{x}\approx \ln 3$
Compute the limit for $b=5$:
$\dfrac{5^{-0.1}-1}{-0.1}\approx 1.4866008$
$\dfrac{5^{-0.01}-1}{-0.01}\approx 1.5965557$
$\dfrac{5^{-0.001}-1}{-0.001}\approx 1.6081435$
$\dfrac{5^{-0.0001}-1}{-0.0001}\approx 1.6093084$
$\dfrac{5^{-0.00001}-1}{-0.00001}\approx 1.609425$
$\dfrac{5^{0.00001}-1}{0.00001}\approx 1.6094509$
$\dfrac{5^{0.0001}-1}{0.0001}\approx 1.6095674$
$\dfrac{5^{0.001}-1}{0.001}\approx 1.6107338$
$\dfrac{5^{0.01}-1}{0.01}\approx 1.6224591$
$\dfrac{5^{0.1}-1}{0.1}\approx 1.7461894$
$\ln 5\approx 1.60943791$
Therefore we got:
$\dfrac{5^{x}-1}{x}\approx 1.6094=\ln 5$
Conjecture:
$\displaystyle\lim_{x\rightarrow 0} \dfrac{b^x-1}{x}=\ln b$
Test the conjecture for $b=7$ and $b=10$:
Compute the limit for $b=5$:
$\dfrac{7^{-0.1}-1}{-0.1}\approx 1.7682875$
$\dfrac{7^{-0.01}-1}{-0.01}\approx 1.9270995$
$\dfrac{7^{-0.001}-1}{-0.001}\approx 1.9440181$
$\dfrac{7^{-0.0001}-1}{-0.0001}\approx 1.9457208$
$\dfrac{7^{-0.00001}-1}{-0.00001}\approx 1.9459291$
$\dfrac{7^{0.00001}-1}{0.00001}\approx 1.9459291$
$\dfrac{7^{0.0001}-1}{0.0001}\approx 1.9460995$
$\dfrac{7^{0.001}-1}{0.001}\approx 1.9478047$
$\dfrac{7^{0.01}-1}{0.01}\approx 1.9649664$
$\dfrac{7^{0.1}-1}{0.1}\approx 2.1481404$
$\displaystyle\lim_{x\rightarrow 0} \dfrac{7^x-1}{x}=1.9459$
$\ln 7\approx 1.9459$
$\Rightarrow \displaystyle\lim_{x\rightarrow 0} \dfrac{7^x-1}{x}=\ln 7\checkmark$
Compute the limit for $b=10$:
$\dfrac{10^{-0.1}-1}{-0.1}\approx 2.0567177$
$\dfrac{10^{-0.01}-1}{-0.01}\approx 2.2762779$
$\dfrac{10^{-0.001}-1}{-0.001}\approx 2.2999362$
$\dfrac{10^{-0.0001}-1}{-0.0001}\approx 2.30232$
$\dfrac{10^{-0.00001}-1}{-0.00001}\approx 2.3025586$
$\dfrac{10^{0.00001}-1}{0.00001}\approx 2.3026116$
$\dfrac{10^{0.0001}-1}{0.0001}\approx 2.3028502$
$\dfrac{10^{0.001}-1}{0.001}\approx 2.3052381$
$\dfrac{10^{0.01}-1}{0.01}\approx 2.3292992$
$\dfrac{10^{0.1}-1}{0.1}\approx 2.5892541$
$\displaystyle\lim_{x\rightarrow 0} \dfrac{10^x-1}{x}=2.3026$
$\ln 10\approx 2.3026$
$\Rightarrow \displaystyle\lim_{x\rightarrow 0} \dfrac{10^x-1}{x}=\ln 10\checkmark$
