Find the limits:?

Question

Hi, Just need a hand solving these any help would be great. showing working out will get the best answer.
Thanks

1)
lim cos( 7 n^−1 ) =
n → 0+

2)
lim cos( π n) =
n → ∞

3)
lim tan(π/n)
n → ∞

Answer 1

lim cos( 7 n^−1 ) =
n → 0+
lim cos( 7 / n ) =
n → 0+
lim 7 / n = infinity
n → 0+
cosinus doesn't have a value for infinity instead it could be anything between -1 to 1 so
lim cos( 7 n^−1 ) doesn't exist
n → 0+

lim cos( π n) =
n → ∞
if n is a odd then cos (π n) = -1 else n will be even and cos (π n) = 1
even in this case the limit doesn't exist because it could be -1 or 1.

lim tan(π/n) =
n → ∞
lim π/n = 0 which gives
n → ∞
lim tan(π/n) = tan 0 = 0
n → ∞

Answer 2

n normally represents a positive integer
In the first case it doesn´t
7/n==> to + infinity and cos (7/n) has NO limit
You can take n= 1/2kpi so 7/n = 14kpi and cos 14kpi is always 1
If you take n= 1(2k+1)pi cos 14kpi+7pi =-1 as k integer==>+infinity
The same for 2)
3) lim tan(pi/n) n==> infinity is 0

Answer 3

3) as n->infinity pi/n ->0
tan(pi/n)->0
answer is 0