If \(\lim_{n\rightarrow \infty}\)(\(\sqrt{n^2-n-1}\) + nα+β)=0 then 8(α + β) is equal to
- 4
- -8
- -4
- 8
The Correct Option is C
Solution and Explanation
\(\lim_{n\rightarrow \infty}\)(\(n^2−n−1\)+nα+β)=0
=\(\lim_{n\rightarrow \infty}\) n[\(\sqrt{1-\frac{1}{n}-\frac{1}{n^2}}\)+α+\(\frac{\beta}{n}\)]=0
∴ α = –1
Now,
\(\lim_{n\rightarrow \infty}\) n[{1−(\(\frac{1}{n}-\frac{1}{n^2}\))}\(^{\frac{1}{2}}\)+\(\frac{\beta }{n}\)−1]=0
\(\lim_{n\rightarrow \infty}\) n(1−\(\frac{1}{2}\)(\(\frac{1}{n}+\frac{1}{n^2}\))+…)+\(\frac{\beta }{n}\)−1=0
⇒ β–\(\frac{1}{2}\)=0
∴β=\(\frac{1}{2}\)
Now,
8(α+β)=8(-\(\frac{1}{2}\))=-4
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Concepts Used:
Limits
A function's limit is a number that a function reaches when its independent variable comes to a certain value. The value (say a) to which the function f(x) approaches casually as the independent variable x approaches casually a given value "A" denoted as f(x) = A.
If limx→a- f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the left of ‘a’. This value is also called the left-hand limit of ‘f’ at a.
If limx→a+ f(x) is the expected value of f when x = a, given the values of ‘f’ near x to the right of ‘a’. This value is also called the right-hand limit of f(x) at a.
If the right-hand and left-hand limits concur, then it is referred to as a common value as the limit of f(x) at x = a and denote it by lim x→a f(x).