## anonymous one year ago State the value of the limit, if it exists lim x→0 sin(x) multiplied with 3x^3+2x^2/x^2 Anybody? :)

1. anonymous

ok this is a big one

2. anonymous

so stay with me

3. anonymous

first we separate the lim

4. anonymous

$\lim_{x \rightarrow 0} (3x+2) \times \lim_{x \rightarrow 0}(\frac{ \sin(x) }{ x ^{2} })$

5. anonymous

i forgot to times x^2 with sin(x)

6. anonymous

ok

7. anonymous

$\lim_{x \rightarrow 0}(\frac{ \sin(x)x ^{2} }{ x ^{2} }) \times \lim_{x \rightarrow 0} (3x+2)$

8. anonymous

ur following the steps right

9. anonymous

wow! :)

10. anonymous

Yep! Think so…

11. anonymous

3x+2 is a polynomial and thus everywhere continuous so $\lim_{x \rightarrow 0} (3x+2)= 2+3(0)=2$

12. anonymous

and the $\lim_{x \rightarrow 0} \frac{ \sin(0) \times 0 }{ 0 } =0$

13. anonymous

$2 \times 0=0$

14. anonymous

got it

15. anonymous

if u have similar denominator like x^2 in ur case u can separate the limit

16. anonymous

Yup! Think so…. :)

17. anonymous

I have to go through it a few times to make sure I understand everything :)

18. anonymous

Thank you so much!

19. anonymous

Any possibilities of some follow-up questions later on?

20. anonymous

Yea sure

21. IrishBoy123

you simply can not say this: $\lim_{x \rightarrow 0}(\frac{ \sin(x)x ^{2} }{ x ^{2} })$ $\implies \lim_{x \rightarrow 0} \frac{ \sin(0) \times 0 }{ 0 } =0$ the whole point of this is that dividing by zero you cannot do, so you check what the function in question does as you approach very close to zero. so: $\lim\limits_{ x→0} \, \, \sin(x) .\frac{3x^3+2x^2}{x^2}$ $=\lim\limits_{ x→0} \, \, \sin(x) .\lim\limits_{ x→0}\frac{3x^3+2x^2}{x^2}$ $=\lim\limits_{ x→0} \, \, \sin(x) .\lim\limits_{ x→0}\frac{\frac{3x^3}{x^2}+\frac{2x^2}{x^2}}{\frac{x^2}{x^2}}$ $=\lim\limits_{ x→0} \, \, \sin(x) .\lim\limits_{ x→0}\frac{3x+2}{1}$ $=0 \times 2 = 0$

22. anonymous