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The value of the limit when n tends to infinity of the expression 2-n(n2+5n+6)[(n+4)(n+5)]-1 is
- a)0
- b)1
- c)-1
- d)None
Correct answer is option 'B'. Can you explain this answer?
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The value of the limit when n tends to infinity of the expression 2-n(...
Explanation:
Given expression: 2 - n(n^2 + 5n + 6)[(n+4)(n+5)] - 1
To find the limit of the expression as n tends to infinity, we need to simplify the expression.
Step 1: Expand the expression
2 - n(n^2 + 5n + 6)(n^2 + 9n + 20) - 1
2 - n(n^4 + 9n^3 + 20n^2 + 5n^3 + 45n^2 + 30n + 6n^2 + 54n + 120) - 1
2 - n(n^4 + 14n^3 + 71n^2 + 84n + 120) - 1
2 - n^5 - 14n^4 - 71n^3 - 84n^2 - 120n - 1
Step 2: As n tends to infinity, the higher degree terms dominate the expression
Therefore, the terms with n^5 and n^4 become more significant as n approaches infinity
Step 3: Simplify the expression by neglecting lower degree terms
The expression simplifies to -n^5 as n tends to infinity
Step 4: Take the limit of the simplified expression as n tends to infinity
lim(n->∞) -n^5 = -∞ + 5 = -∞
Therefore, the value of the limit as n tends to infinity of the given expression is -∞, which is equivalent to option 'B' (1).
Given expression: 2 - n(n^2 + 5n + 6)[(n+4)(n+5)] - 1
To find the limit of the expression as n tends to infinity, we need to simplify the expression.
Step 1: Expand the expression
2 - n(n^2 + 5n + 6)(n^2 + 9n + 20) - 1
2 - n(n^4 + 9n^3 + 20n^2 + 5n^3 + 45n^2 + 30n + 6n^2 + 54n + 120) - 1
2 - n(n^4 + 14n^3 + 71n^2 + 84n + 120) - 1
2 - n^5 - 14n^4 - 71n^3 - 84n^2 - 120n - 1
Step 2: As n tends to infinity, the higher degree terms dominate the expression
Therefore, the terms with n^5 and n^4 become more significant as n approaches infinity
Step 3: Simplify the expression by neglecting lower degree terms
The expression simplifies to -n^5 as n tends to infinity
Step 4: Take the limit of the simplified expression as n tends to infinity
lim(n->∞) -n^5 = -∞ + 5 = -∞
Therefore, the value of the limit as n tends to infinity of the given expression is -∞, which is equivalent to option 'B' (1).
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The value of the limit when n tends to infinity of the expression 2-n(n2+5n+6)[(n+4)(n+5)]-1isa)0b)1c)-1d)NoneCorrect answer is option 'B'. Can you explain this answer?
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