find the following limit
lim
n->infinity 2+5n+4n^4/2n
show steps please
when n gets big, 4 n^4/ 2n dominates so 2 n^3
so its no limit which would just be infinity right?
yes
To find the given limit, we can apply the concept of limit rules and simplify the expression before evaluating it.
Given:
lim(n->infinity) (2 + 5n + 4n^4) / (2n)
Step 1: Simplify the expression:
Divide every term in the numerator by 2n:
= (2/2n + 5n/2n + 4n^4/2n)
Step 2: Further simplify:
= (1/n + 5/2 + 2n^3)
Step 3: Take the limit as n approaches infinity:
lim(n->infinity) (1/n + 5/2 + 2n^3)
Since we are taking the limit as n approaches infinity, the term 1/n becomes negligible because as n gets larger and larger, 1/n approaches zero.
Thus, we can disregard 1/n in the expression, leaving us with:
lim(n->infinity) (5/2 + 2n^3)
Step 4: Finally, evaluate the limit:
As n approaches infinity, 5/2 + 2n^3 will also approach infinity.
Therefore, the limit is infinity.