Question

More limits :)

Answer 1

#7

Answer 2

\[
|x|
=
\begin{cases}
x \ \text{ if } \ x \ge 0\\
-x \ \text{ if } \ x < 0\\
\end{cases}
\]

\[
|\color{red}{x}|
=
\begin{cases}
(\color{red}{x}) \ \text{ if } \ (\color{red}{x}) \ge 0\\
-(\color{red}{x}) \ \text{ if } \ (\color{red}{x}) < 0\\
\end{cases}
\]

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

\[
|2x+1|
=
\begin{cases}
2x+1 \ \text{ if } \ x \ge -\frac{1}{2}\\
-2x-1 \ \text{ if } \ x < -\frac{1}{2}\\
\end{cases}
\]

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)

Answer 3

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)

Answer 4

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

Answer 5

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

Answer 6

ah k gotcha.

Answer 7

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

Answer 8

correct

Answer 9

Just a sec.

Answer 10

ok

Answer 11

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

Answer 12

close, there's an x missing

Answer 13

Right on the second 2, sorry xD

Answer 14

But that al simplifies to
4x/x?

Answer 15

which of course = 4

Answer 16

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

Answer 17

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

Answer 18

It should be -4. The graph confirms it

https://www.desmos.com/calculator/4uqoaaryhi

Definitely an odd and interesting graph

Answer 19

haha looks...cool?

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

oh! lovely!