More limits :)

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More limits :)

Mathematics
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#7
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\[ |x| = \begin{cases} x \ \text{ if } \ x \ge 0\\ -x \ \text{ if } \ x < 0\\ \end{cases} \] \[ |\color{red}{x}| = \begin{cases} (\color{red}{x}) \ \text{ if } \ (\color{red}{x}) \ge 0\\ -(\color{red}{x}) \ \text{ if } \ (\color{red}{x}) < 0\\ \end{cases} \] \[ |\color{red}{2x+1}| = \begin{cases} (\color{red}{2x+1}) \ \text{ if } \ (\color{red}{2x+1}) \ge 0\\ -(\color{red}{2x+1}) \ \text{ if } \ (\color{red}{2x+1}) < 0\\ \end{cases} \] \[ |2x+1| = \begin{cases} 2x+1 \ \text{ if } \ x \ge -\frac{1}{2}\\ -2x-1 \ \text{ if } \ x < -\frac{1}{2}\\ \end{cases} \] 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)
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)

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How did you come up with the 1/2 's ?
solving 2x+1 >= 0, 2x+1 < 0, etc etc. The restrictions placed on each piece of the piecewise function
ah k gotcha.
So now we replace the values in the original with the ones you've just come up with.
correct
Just a sec.
ok
So it's [(2x+1)-(-2+1)]/x
close, there's an x missing
Right on the second 2, sorry xD
But that al simplifies to 4x/x?
which of course = 4
Sorry you should have this \[ \lim_{x\to 0} \frac{|2x-1|-|2x+1|}{x} = \lim_{x\to 0} \frac{(-2x+1)-(2x+1)}{x} \]
you somehow mixed up |2x-1| and |2x+1|
It should be -4. The graph confirms it https://www.desmos.com/calculator/4uqoaaryhi Definitely an odd and interesting graph
haha looks...cool?
okay, I got it right now :) so the entire limit = - 4
correct, as x gets closer and closer to 0 (from either side) the value of y gets closer and closer to -4
Fabulous. Thank you so much. So at the end I would just rewrite the original limit and then put it all equals -4? correct?
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
oh! lovely!

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