What is the value of this expression when n approaches infinity?
4 −(4/n)+(5/n)+(15/3n^2) *its all one equation*
when n - infinity then 1/n tends to 0 so 4
To find the value of this expression as n approaches infinity, we need to determine the limit of the expression as n becomes larger and larger.
Let's break down the given expression and simplify it step by step:
Expression: 4 − (4/n) + (5/n) + (15/3n^2)
Step 1: Simplify the expression within the parentheses:
4 − (4/n) + (5/n) + (15/3n^2)
= 4 − 4/n + 5/n + 15/(3n^2)
Step 2: Combine like terms:
= (4 + 5)/n - 4/n + 15/(3n^2)
Step 3: Find the least common denominator (LCD) to add the fractions:
LCD = n * 3n^2 = 3n^3
Step 4: Rewrite each fraction with the LCD:
= (9n + 15n^2)/(n * 3n^2) - 4/n + 15/(3n^2)
= (9n + 15n^2)/(3n^3) - 4/n + 15/(3n^2)
Step 5: Simplify each term:
As n approaches infinity, the terms, 4/n and 15/(3n^2), will both tend to zero, because the denominator becomes infinitely large while the numerator remains constant.
Therefore, in the expression, the limits of these terms are:
lim(4/n) = 0 as n → ∞
lim(15/(3n^2)) = 0 as n → ∞
Now, let's focus on the first term:
lim((9n + 15n^2)/(3n^3)) as n → ∞
Dividing both the numerator and denominator by n^3, we get:
lim((9/n^2 + 15n)/(3n^2)) as n → ∞
Now, as n approaches infinity, the terms (9/n^2) and (15n/(3n^2)) will both tend to zero, because the denominator becomes infinitely large while the numerator remains constant.
Therefore, in this term, the limits of these fractions are:
lim(9/n^2) = 0 as n → ∞
lim((15n)/(3n^2)) = 0 as n → ∞
Finally, if we add up all the terms with their limits:
lim(4 − (4/n) + (5/n) + (15/3n^2)) as n → ∞
= 0 + 0 + 0
= 0
Hence, as n approaches infinity, the value of the given expression is 0.