by SuperAwesomeCaptain McFluffyPa
Last Updated October 09, 2019 15:20 PM

This is a tricky problem. Can anyone help me with the procedure and answer?

Evaluate $$ \lim_{h\to 0} \left( \frac{f(x+hx)}{f(x)}\right)^{1/h}, \text{for }f(x)=x. $$

https://i.stack.imgur.com/QGBNf.png

Note that you are given $f(x)=x$. With this piece of information, we can replace the functions in the given equation with their respective algebraic representations. For instance, the denominator would simply be $x$. What would the numerator look like? Now, as $h\rightarrow 0$, consider what the value inside the parenthesis tends towards, and similarly for the exponent.

**Hint** Taking the logarithm, you have to calculate
$$\lim_{h\to 0} \frac{\ln(f(x+hx))- \ln(f(x))}{h}$$

That is the definition of the derivative of $\ln( f(x))$.

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