Limits of three functions

Could you help me calculate the following limits:

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

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

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

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

Re: Limits of three functions

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

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

Re: Limits of three functions

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Originally Posted by

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

**What an absolutely useless reply!**

Can you explain any of that? I doubt it.

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Re: Limits of three functions

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 $\displaystyle log_{10}10$.

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

Re: Limits of three functions

Quote:

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, $\displaystyle \left[\frac{1}{x}\right]$, is NOT the "floor function", it is just a complicated way of writing $\displaystyle \frac{1}{x}$.

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

Quote:

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 $\displaystyle log_{10}10$.

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