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Chapter 7: Problem 35

Evaluate \(\lim _{n \rightarrow \infty} n^{2}\left(1-\cos \frac{1}{n}\right)\).

Short Answer

Expert verified
The limit \(\lim_{n\rightarrow\infty}n^2(1-\cos{\frac{1}{n}})\) is equal to 2.

Step by step solution

01

Rewrite the expression using sine function

First, we rewrite the given expression using the identity: \(1-\cos x = 2\sin^2{\frac{x}{2}}\). So, our limit becomes: \[\lim _{n \rightarrow \infty} n^{2}\left(2\sin^2{\frac{1}{2n}}\right).\]
02

Simplify the expression

Now, we can simplify this expression by dividing both the numerator and the denominator by \(n^2\): \[\lim _{n \rightarrow \infty}\frac{2\sin^2 {\frac{1}{2n}}}{\left(\frac{1}{2n}\right)^2}.\]
03

Apply the limit equivalence

Now, we can apply the limit equivalence mentioned in the analysis: \[\lim _{x \rightarrow 0} \frac{\sin^2 x}{x^2} = 1.\] Let \(x = \frac{1}{2n}\). As \(n\to\infty\), \(x\to 0\). Thus, our limit becomes: \[\lim _{x \rightarrow 0} \frac{2\sin^2 {x}}{x^2}.\] Using the limit equivalence, we get: \[\lim _{x \rightarrow 0} 2\lim _{x \rightarrow 0} \frac{\sin^2 {x}}{x^2}\] The limit of the expression is: \[2\lim_{x\to0}\frac{\sin^2 x}{x^2} = 2 \times 1.\]
04

Calculate the value of the limit

Finally, we calculate the value of the limit: \[2 \times 1 = 2.\] Thus, the limit \(\lim_{n\rightarrow\infty}n^2(1-\cos{\frac{1}{n}})\) is equal to 2.

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