## Babynini one year ago More limits :)

1. Babynini

#7

2. jim_thompson5910

$|x| = \begin{cases} x \ \text{ if } \ x \ge 0\\ -x \ \text{ if } \ x < 0\\ \end{cases}$ $|\color{red}{x}| = \begin{cases} (\color{red}{x}) \ \text{ if } \ (\color{red}{x}) \ge 0\\ -(\color{red}{x}) \ \text{ if } \ (\color{red}{x}) < 0\\ \end{cases}$ $|\color{red}{2x+1}| = \begin{cases} (\color{red}{2x+1}) \ \text{ if } \ (\color{red}{2x+1}) \ge 0\\ -(\color{red}{2x+1}) \ \text{ if } \ (\color{red}{2x+1}) < 0\\ \end{cases}$ $|2x+1| = \begin{cases} 2x+1 \ \text{ if } \ x \ge -\frac{1}{2}\\ -2x-1 \ \text{ if } \ x < -\frac{1}{2}\\ \end{cases}$ based on the last definition above, we can replace the |2x+1| with just 2x+1 because x is getting closer and closer to 0 (0 makes x >= -1/2 true)

3. jim_thompson5910

Similarly, $|2x-1| = \begin{cases} 2x-1 \ \text{ if } \ x \ge \frac{1}{2}\\ -2x+1 \ \text{ if } \ x < \frac{1}{2}\\ \end{cases}$ So |2x-1| can be replaced with -2x+1 (x=0 makes x < 1/2 true)

4. Babynini

How did you come up with the 1/2 's ?

5. jim_thompson5910

solving 2x+1 >= 0, 2x+1 < 0, etc etc. The restrictions placed on each piece of the piecewise function

6. Babynini

ah k gotcha.

7. Babynini

So now we replace the values in the original with the ones you've just come up with.

8. jim_thompson5910

correct

9. Babynini

Just a sec.

10. jim_thompson5910

ok

11. Babynini

So it's [(2x+1)-(-2+1)]/x

12. jim_thompson5910

close, there's an x missing

13. Babynini

Right on the second 2, sorry xD

14. Babynini

But that al simplifies to 4x/x?

15. Babynini

which of course = 4

16. jim_thompson5910

Sorry you should have this $\lim_{x\to 0} \frac{|2x-1|-|2x+1|}{x} = \lim_{x\to 0} \frac{(-2x+1)-(2x+1)}{x}$

17. jim_thompson5910

you somehow mixed up |2x-1| and |2x+1|

18. jim_thompson5910

It should be -4. The graph confirms it https://www.desmos.com/calculator/4uqoaaryhi Definitely an odd and interesting graph

19. Babynini

haha looks...cool?

20. Babynini

okay, I got it right now :) so the entire limit = - 4

21. jim_thompson5910

correct, as x gets closer and closer to 0 (from either side) the value of y gets closer and closer to -4

22. Babynini

Fabulous. Thank you so much. So at the end I would just rewrite the original limit and then put it all equals -4? correct?

23. jim_thompson5910

yeah after you simplify you'll get -4x/x = -4 then you just apply the limit $\Large \lim_{x\to 0}(-4)$ to get -4 itself

24. Babynini

oh! lovely!