+37 Hey. Currently trying to find the limit as n goes to infinity. I know it converges because I've done this before, but my question is if it converges to -0 or positive 0. This isn't something the question is asking but for my peace of mind.
\(\lim_{x\rightarrow inf} (-7)^n/(n\sqrt{n})7^n\)
Find the following limit:
lim_(n->∞) (-1)^n n^(-3/2)
Applying the quotient rule, write lim_(n->∞) (-1)^n/n^(3/2) as (lim_(n->∞) (-1)^n)/(lim_(n->∞) n^(3/2)):
(lim_(n->∞) (-1)^n)/(lim_(n->∞) n^(3/2))
lim_(n->∞) (-1)^n = lim_(n->∞) e^(log((-1)^n)):
lim_(n->∞) e^(log((-1)^n))/(lim_(n->∞) n^(3/2))
e^(log((-1)^n)) = exp(i n π):
(lim_(n->∞) exp(i π n))/(lim_(n->∞) n^(3/2))
lim_(n->∞) e^(i π n) = e^(lim_(n->∞) i π n):
e^(lim_(n->∞) i π n)/(lim_(n->∞) n^(3/2))
Applying the product rule, write lim_(n->∞) i π n as i π (lim_(n->∞) n):
e^(i π lim_(n->∞) n)/(lim_(n->∞) n^(3/2))
lim_(n->∞) n = ∞:
e^(i π ∞)/(lim_(n->∞) n^(3/2))
i π ∞ = i ∞:
e^(i ∞)/(lim_(n->∞) n^(3/2))
Using the power rule, write lim_(n->∞) n^(3/2) as (lim_(n->∞) n)^(3/2):
(undefined)/lim_(n->∞) n^(3/2)
lim_(n->∞) n = ∞:
(undefined)/∞^(3/2)
∞^(3/2) = ∞:
Answer: | (undefined)/∞
