How do I solve this particularly tricky limit problem?

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

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

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

Tags : calculus limits

Answers 2

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

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