two question about limit. help please!

[asomeras]

\(\displaystyle \mbox{Problem 1.12. Evaluate the limit }\, \displaystyle{\lim_{x\, \rightarrow\, 0}\, x^4\, \cos\left(\frac{2}{x}\right)}\)

\(\displaystyle \mbox{Problem 1.13. Evaluate the limit }\, \displaystyle{\lim_{x\, \rightarrow\, 0}\, x\, \sin\left(\frac{1}{x}\right)}\)

can someone explain how can i solve these questions?

 

[pka]

\(\displaystyle \mbox{Problem 1.12. Evaluate the limit }\, \displaystyle{\lim_{x\, \rightarrow\, 0}\, x^4\, \cos\left(\frac{2}{x}\right)}\)

\(\displaystyle \mbox{Problem 1.13. Evaluate the limit }\, \displaystyle{\lim_{x\, \rightarrow\, 0}\, x\, \sin\left(\frac{1}{x}\right)}\)

can someone explain how can i solve these questions?

Is it true that \(\displaystyle - {x^4} \le {x^4}\cos \left( {\dfrac{2}{x}} \right) \le {x^4}\quad ?\)

Is it true that \(\displaystyle x\sin \left( {{x^{ - 1}}} \right) = \dfrac{{\sin \left( {{x^{ - 1}}} \right)}}{{{x^{ - 1}}}}\quad ?\)

 

[asomeras]

Is it true that \(\displaystyle - {x^4} \le {x^4}\cos \left( {\dfrac{2}{x}} \right) \le {x^4}\quad ?\)

Is it true that \(\displaystyle x\sin \left( {{x^{ - 1}}} \right) = \dfrac{{\sin \left( {{x^{ - 1}}} \right)}}{{{x^{ - 1}}}}\quad ?\)

i do not understand what do you mean :?

 

[pka]

i do not understand what do you mean

Then you do not have the necessary background to do these questions.
If I were I would seek out a tutor.

 

[Ishuda]

\(\displaystyle \mbox{Problem 1.12. Evaluate the limit }\, \displaystyle{\lim_{x\, \rightarrow\, 0}\, x^4\, \cos\left(\frac{2}{x}\right)}\)

\(\displaystyle \mbox{Problem 1.13. Evaluate the limit }\, \displaystyle{\lim_{x\, \rightarrow\, 0}\, x\, \sin\left(\frac{1}{x}\right)}\)

can someone explain how can i solve these questions?

If we have the limit of the product of two functions, and one is bounded and the other goes to zero, then the limit of the product is zero. Are you allowed to use this fact? That is, has it been proved in class or has your instructor given you this? If so consider that the sine and cosine are bounded. If you can't use it, can you prove it so that you can use it?