Let (an) be a sequence of positive real numbers such that lim an 1/an =1/4.


We need to find the limit of (e^an² an)/4.



We are given that lim an 1/an =1/4. This can be written as:

lim (an - 1/an) = 1/4


We can rewrite the expression as:

lim (an - 1) = 1/4 * lim an

Taking the reciprocal of both sides gives:

lim (1/(an - 1)) = 4 * lim (1/an)


Since we are given that lim an 1/an = 1/4, we can substitute this into the equation:

lim (1/(an - 1)) = 4 * (1/4)

Simplifying further, we get:

lim (1/(an - 1)) = 1


Since we have the limit of (1/(an - 1)), we can use the limit property of exponential functions to find the limit of (e^an² an)/4. This property states that if lim f(x) = L, then lim e^f(x) = e^L.

Applying this property, we have:

lim e^(an - 1) = e^1


We can rewrite the expression as:

lim (e^an * e^(-1)) = e

Since e^(-1) is a constant, we can take it outside of the limit:

e^(-1) * lim e^an = e

We can then solve for the limit:

lim e^an = e / e^(-1) = e^2


Finally, we substitute the value of lim e^an into the original expression:

lim (e^an² an)/4 = lim (e^an * e^an * an)/4 = (e^2 * e^2 * 1)/4 = e^4 / 4

Therefore, the limit of (e^an² an)/4 is e^4 / 4.