The discussion revolves around evaluating the limit of the expression (1 + a/x)^x as x approaches infinity. Participants are exploring the implications of the exponent and the behavior of the expression as x increases.
There are multiple lines of reasoning being explored, with some participants providing detailed approaches involving logarithmic transformations and L'Hôpital's Rule, while others question the necessity of such methods. The discussion reflects a mix of interpretations regarding the limit's evaluation.
Participants are navigating through various definitions of the mathematical constant e and the implications of indeterminate forms that arise in the limit evaluation. There is a recognition of the complexity of the problem, as indicated by the differing opinions on the length and depth of the solution process.
DavidWhitbeck said:Caesius, have you seen all definitions of e?
QUOTE]
No one has seen all definitions of e as there are an infinite number of them.
rocomath said:\lim_{x\rightarrow\infty}\left(1+\frac a x\right)^x
Set it equal to y, take the natural log of both sides.
y=\left(1+\frac a x\right)^x
\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}x\ln\left(1+\frac a x\right)
\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}x\ln\left(1+\frac a x\right)
Now you have an Indeterminate form of \infty\cdot0
Keep solving till you can apply L'Hopital's Rule on the right side, then you will have "solved for \ln y, so use algebra to solve for "y" and you're pretty much done.
mubashirmansoor said:Thats a great solution but I think no need to go that long;
as x approaches infinity "(a/x)" equates 0 .
Hence : 1+0 = 1
Indeterminate Power of form: 1^{\infty} so yes it is necessary.mubashirmansoor said:Thats a great solution but I think no need to go that long;
as x approaches infinity "(a/x)" equates 0 .
Hence : 1+0 = 1