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Deduce a limit relation using the derivative of ecx

by RoRi

Consider the function f(x) = e^{cx} for a constant c. First, show that f'(0) = c and then prove

    \[ \lim_{x \to 0} \frac{e^{cx}-1}{x} = c. \]

Proof. If f(x) = e^{cx} then we have

    \[ f'(x) = ce^{cx} \quad \implies \quad f'(0) = ce^0 = c. \]

From the definition of the derivative we also know

    \begin{align*} &&f'(0) &= \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} \\ \implies && c&= \lim_{h \to 0} \frac{e^{ch} - 1}{h} \\ \implies && c &= \lim_{x \to 0} \frac{e^{cx}- 1}{x}. \qquad \blacksquare \end{align*}

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