We have an answer to "I'd love your help with finding the following..."

High School
Calculus and Analysis
Calculus 1
Limits and continuity

Jasiah Vincent

Answered question

2022-04-06

I'd love your help with finding the following limit:

lim_{n \to \infty} \cos (π \sqrt{n^{2} - n})

Answer & Explanation

firenzesunzc65

Beginner2022-04-07Added 16 answers

$\cos (π \sqrt{n^{2} - n}) = {(- 1)}^{n} \cos (π (\sqrt{n^{2} - n} - n))$
$= {(- 1)}^{n} \cos π \frac{- n}{\sqrt{n^{2} - n} + n}$
$= {(- 1)}^{n} \cos π \frac{1}{\sqrt{1 - \frac{1}{n}} + 1}$
hence $| \cos (π \sqrt{n^{2} - n}) | = | \cos (π \frac{1}{\sqrt{1 - \frac{1}{n}} + 1}) |$ . Since $lim_{n \to + \infty} π \frac{1}{\sqrt{1 - \frac{1}{n}} + 1} = \frac{π}{2}$ , the $\cos$ is continuous and $\cos \frac{π}{2} = 0$ we conclude that the limit is 0

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function
$f (x, y) = x^{3} - 6 x y + 8 y^{3}$
$\frac{1}{\sec (x)}$ in derivative?
What is the derivative of $\ln (x + 1)$ ?
What is the limit of $e^{- x}$ as $x \to \infty$ ?
What is the derivative of $f (x) = 5^{\ln x}$ ?
What is the derivative of $e^{- 2 x}$ ?
How to find $lim \frac{e^{t} - 1}{t}$ as $t \to 0$ using l'Hospital's Rule?
What is the integral of $\sqrt{9 - x^{2}}$ ?
What is the derivative of $f (x) = \ln [x^{9} {(x + 3)}^{6} {(x^{2} + 7)}^{5}]$ ?
What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?
How to find the derivative of $y = e^{5 x}$ ?
How to evaluate the limit $\frac{\sin (5 x)}{x}$ as x approaches 0?
How to find derivatives of parametric functions?
What is the antiderivative of $- 5 e^{x - 1}$ ?
How to evaluate: indefinite integral $\frac{1 + x}{1 + x^{2}} d x$ ?

Didn't find what you were looking for?