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


Find the limit of (e^(an^2) * an)/4 as n approaches infinity.




Let's rewrite the given expression by multiplying and dividing by (1/an):

(e^(an^2) * an)/(4 * (1/an))

Simplifying this expression, we get:

(e^(an^2) * an^2)/4


Recall the definition of the exponential function e^x:

e^x = lim (n→∞) (1 + x/n)^n

Let's rewrite e^(an^2) using this definition:

e^(an^2) = lim (n→∞) (1 + (an^2)/n)^n


Now, let's substitute this expression back into our original limit expression:

lim (n→∞) (e^(an^2) * an^2)/4

= lim (n→∞) [(1 + (an^2)/n)^n * an^2]/4

= lim (n→∞) [(1 + a^2 * (n^2)/n)^n * an^2]/4

= lim (n→∞) [(1 + a^2 * n)^n * an^2]/4


Now, we can simplify the expression inside the limit:

lim (n→∞) [(1 + a^2 * n)^n * an^2]/4

= lim (n→∞) [(1 + a^2 * n)^n] * lim (n→∞) [an^2]/4

The first limit is of the form (1 + ∞)^∞, which is an indeterminate form. We can rewrite it as e^(∞ * ln(1 + a^2 * n)). As n approaches infinity, ln(1 + a^2 * n) approaches infinity, and the limit becomes e^∞, which is equal to infinity.

The second limit, lim (n→∞) [an^2]/4, simplifies to infinity/4 = infinity.

Thus, the overall limit:

lim (n→∞) [(1 + a^2 * n)^n * an^2]/4

is equal to infinity multiplied by infinity, which is still infinity.

Therefore, the limit of (e^(an^2) * an)/4 as n approaches infinity is infinity.