# Thread: Limits of three functions

1. ## Limits of three functions

Could you help me calculate the following limits:

$\lim_{x \to 0} x \left[ \frac{1}{x} \right]$

$\lim_{x\to 0} \frac{1-\cos x \cdot \sqrt{\cos2x} }{x^2}$

$\lim_{x\to 10} \frac{\log _{10}(x) - 1}{x-10}$

As to the last one I thought I could use $\lim\frac{log _{a}(1+\alpha)}{\alpha} = \log_ae$ but it wouldn't work

2. ## Re: Limits of three functions

Is the first limit equal to 1, the second to 1,5 and the third one to $0,1\log_{10}e$ ?

3. ## Re: Limits of three functions

I just did the second one I think it's 3/2. I may have made some computational errors but you simply use l'Hopital's rule twice because you have the form 0/0 twice

4. ## Re: Limits of three functions

Originally Posted by wilhelm
Is the first limit equal to 1, the second to 1,5 and the third one to $0,1\log_{10}e$ ?
Can you explain any of that? I doubt it.

6. ## Re: Limits of three functions

@Plato But of course I can. To compute the first limit I simply used the squeeze theorem and floor function inequality. As to the second one I could use the de l'Hospital theorem but I used simple trigonometric transformations instead. And when it comes to the third one I just noticed that the limit is equal to derivative of $log_{10}10$.

I posted the answers simply because I didn't feel like writing the whole solutions. I'm sorry.

7. ## Re: Limits of three functions

Originally Posted by wilhelm
@Plato But of course I can. To compute the first limit I simply used the squeeze theorem and floor function inequality.
It would have helped if you had said this problem involved the "floor function"! What you wrote, $\left[\frac{1}{x}\right]$, is NOT the "floor function", it is just a complicated way of writing $\frac{1}{x}$.

If you wanted the floor function, write it as $\lfloor x \rfloor$.

As to the second one I could use the de l'Hospital theorem but I used simple trigonometric transformations instead. And when it comes to the third one I just noticed that the limit is equal to derivative of $log_{10}10$.

I posted the answers simply because I didn't feel like writing the whole solutions. I'm sorry.