-1

$$\lim_{x \to \infty} \frac{\cos x}{\cos x}$$

I think that $\cos x$ does not exist if $x\to\infty$. But $\dfrac{\cos x }{\cos x}$. I am confused. Any help will be appreciated.

Joe
  • 19,636
Neetu yadav
  • 13
  • 6
  • Limit of a constant function $f(x)=c$ is just $c$. $\frac{cos(x)}{cos(x)}$ is $1$ for all $x\in\mathbb{R}$ – Mr.Gandalf Sauron Jul 27 '21 at 16:53
  • 4
    @ArghyadeepChatterjee How is $\frac{\cos(x)}{\cos(x)}$ constant? What is its value at $x = \frac{\pi}{2}$? – Good Morning Captain Jul 27 '21 at 16:55
  • 1
    $\infty$ is an accumulation point of the domain of the function $\frac{\cos}{\cos}$, meaning that according to the more general definition of the limit, $\lim_{x \to \infty}\frac{\cos x}{\cos x}=1$. – Joe Jul 27 '21 at 17:01

1 Answers1

3

That fraction has the value $1$ except where $\cos$ has a root. At those places it's undefined, though it would be reasonable to define it as $1$. Whether the limit exists depends on what you want to do at those points. If the limit does exist it will be $1$ since it's the limit of a constant function.

In any case trying to evaluate it as the quotient of the limit of $\cos$ makes no sense (as you note in the question).

Ethan Bolker
  • 95,224
  • 7
  • 108
  • 199