Question

Finding the two-sided limit of the following general form:
\[ \lim_{x→0} (1 − ax)^{1/x}\]
What's the trick here? A medal will be given for showing the steps clearly :-)

Answer 1

\[\large y=(1-ax)^{\frac{1}{x}} \]
\[\large lny=\frac{1}{x}ln(1-ax) \]
now take the limit using L'Hopital's rule.

Answer 2

The trick here is to write \[ (1-ax)^{1/x}=e ^{ \ln(1-ax)/x}\]

Apply l'hopital's rule to the exponent as x goes to zero

ln(1-ax)/ x ---> -a/(1-ax) ----> -a

so that

\[\[ (1-ax)^{1/x}=e ^{ -a}\]

Answer 3

Gj both of you and ty :-)