- #1

- 10

- 0

**why does it work only when h tends to zero????**

[itex]\hat{}\[\begin{array}{l}

f(x + h) = xf(x)\\

Ef(x) = xf(x)\\

E = x\\

\ln E = hD\\

\ln x = hD\\

f(x) = y\\

y\ln x = hDy\\

y\ln x = h\frac{{dy}}{{dx}}\\

\int {} \ln xdx = h\int {} dy/y\\

x\log x/e = h\ln y + \ln c\\

x\log x/e = h\ln y/C\\

(x/h)\ln x/e = \ln y/C\\

C(x/e)\frac{{x/h}}{1} = y = f(x)\\

f(x + h) = C((x + h)/e)\frac{{(x + h)/h}}{1}\\

f(x + h)/f(x) = (((x + h)/e)\frac{{(x + h)/h}}{1})/(x/e)\frac{{x/h}}{1}\\

f(x + h)/f(x) = ((x + h)/x)\frac{{x/h}}{1})*((x + h)/e)\\

f(x + h)/f(x) = (1 + h/x)\frac{{x/h}}{1}*((x + h)/e)\\

\mathop {\lim }\limits_{h \to 0} f(x + h)/f(x) = \mathop {\lim }\limits_{h \to 0} e*((x + h)/e)\\

\mathop {\lim }\limits_{h \to 0} f(x + h)/f(x) = \mathop {\lim }\limits_{h \to 0} (x + h)\\

\mathop {\lim }\limits_{h \to 0} f(x + h)/f(x) = x\\

\\

\end{array}\][/itex]