I am struggling in calculating the limit of the following function
When x approaching zero from the right (0+),
lim (lnx/e^2x)
The problem I have here is how to handle lnx because lnx is not defined at 0
Please help and thanks!
I am struggling in calculating the limit of the following function
When x approaching zero from the right (0+),
lim (lnx/e^2x)
The problem I have here is how to handle lnx because lnx is not defined at 0
Please help and thanks!
First of all, recall that ln(x) tends towards -infinity for x->0+.
For x<1, we have e^(2x) < e^2, hence 1/e^(2x) > 1/e^2.
For x<1, ln(x) is negative, so multiplying both sides of the inequality by ln(x) gives ln(x)/e^(2x) < ln(x)/e^2.
And then you use my initial reminder, which says that the right hand side of this last inequality will tend to -infinity, and hence so will the left hand side.
The limit of a quotient is the quotient of the limits,
provided that the limit of the denominator is not zero.
$\displaystyle \displaystyle\lim_{x \to 0+}\frac{log_ex}{e^{2x}}=k\Rightarrow\frac{\displa ystyle\lim_{x \to 0+}log_ex}{\displaystyle\lim_{x \to 0+}e^{2x}}=k$
$\displaystyle e^0=1$
$\displaystyle \displaystyle\lim_{x \to 0+}log_ex=k\Rightarrow\lim_{x \to 0+}x=e^k$
hence $\displaystyle k\rightarrow\ -\infty$
so there is no limit.
That's not a bad "heuristic" way of doing it but I suspect many teachers would reject it as a valid argument. "The limit of a quotient is the quotient of the limits, provided that the limit of the denominator is not zero" and both limits exist. In this case, the limit of the numerator does not exist. Saying that a limit "is infinity" or "is negative infinity" is just saying the limits do not exist for a particular reason. Happy Joe's argument is better.