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Question Number 185508 by mathlove last updated on 23/Jan/23

$$\underset{x\to 2}{\lim }\frac{5^x-25}{x-2}=?$$$$without{H}^{\prime }Lroule$$

Answered by Ar Brandon last updated on 23/Jan/23

$$\underset{x\to 2}{\lim }\frac{5^x-25}{x-2}=\underset{t\to 0}{\lim }\frac{5^{t+2}-25}{t}=\underset{t\to 0}{\lim }\frac{25(5^t-1)}{t}$$$$=25\underset{t\to 0}{\lim }\frac{e^{t\ln 5}-1}{t}=25\underset{t\to 0}{\lim }\frac{(1+t\ln 5)-1}{t}$$$$=25\underset{t\to 0}{\lim }\frac{t\ln 5}{t}=25\underset{t\to 0}{\lim }\left(\ln 5\right)=25\ln \left(5\right)$$

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