## Calculus (3rd Edition)

$\displaystyle\lim_{x\rightarrow 0} \dfrac{3^x-1}{x}=\ln 3$ $\displaystyle\lim_{x\rightarrow 0} \dfrac{5^x-1}{x}=\ln 5$ $\displaystyle\lim_{x\rightarrow 0} \dfrac{b^x-1}{x}=\ln b$
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 $b=10$: $\dfrac{10^{-0.1}-1}{-0.1}\approx 2.0567177$ $\dfrac{10^{-0.01}-1}{-0.01}\approx 2.2762779$ $\dfrac{10^{-0.001}-1}{-0.001}\approx 2.2999362$ $\dfrac{10^{-0.0001}-1}{-0.0001}\approx 2.30232$ $\dfrac{10^{-0.00001}-1}{-0.00001}\approx 2.3025586$ $\dfrac{10^{0.00001}-1}{0.00001}\approx 2.3026116$ $\dfrac{10^{0.0001}-1}{0.0001}\approx 2.3028502$ $\dfrac{10^{0.001}-1}{0.001}\approx 2.3052381$ $\dfrac{10^{0.01}-1}{0.01}\approx 2.3292992$ $\dfrac{10^{0.1}-1}{0.1}\approx 2.5892541$ $\displaystyle\lim_{x\rightarrow 0} \dfrac{10^x-1}{x}=2.3026$ $\ln 10\approx 2.3026$ $\Rightarrow \displaystyle\lim_{x\rightarrow 0} \dfrac{10^x-1}{x}=\ln 10\checkmark$