How do I solve this particularly tricky limit problem?

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

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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.

JJ Hoo
October 09, 2019 15:14 PM

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 $$\ln( f(x))$$.

N. S.
October 09, 2019 15:16 PM

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Updated November 10, 2017 20:20 PM