- jamiebookeater

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- anonymous

##### 1 Attachment

- asnaseer

have you attempted to sketch the curve first?

- anonymous

|dw:1375034315072:dw|

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## More answers

- asnaseer

which part are you stuck on?

- anonymous

how do i sketch if i dont know the points

- anonymous

all i see are x+1, x <> 2 then im stuck

- anonymous

i can't use calculator.

- asnaseer

ok - lets break it down into smaller steps

- asnaseer

first, you are told that:\[f(x)=x+1\]if \(x\le2\)
agreed?

- anonymous

yes

- asnaseer

so lets sketch this...

- anonymous

so for a) since it;s coming from the left side it must be negative... and we use the top function, correct?

- anonymous

Its already been sketched btw

- asnaseer

|dw:1375034583963:dw|

- anonymous

ok how did you sketch that ? :(

- asnaseer

I just sketched y=x+1
got 2 points on it:
when x=0, y=1
when x=2, y=3
and then just drew a line

- asnaseer

we are told that this is only for x less than or equal to 2.
so we need to CHOP the curve at x=2 and draw a filled-in dot at that point.
it needs to be filled-in because it INCLUDES the value at x=2

- asnaseer

so the actual line should look like this:

- asnaseer

does that make sense?

- anonymous

ohhhhh okay :)

- asnaseer

|dw:1375034970995:dw|

- anonymous

so when are the times that we fill in the dots?

- anonymous

and don't fill in?

- asnaseer

you always fill-in the dots if the point INCLUDES the value - so all these would result in filling in the dot:\[x\le2\]\[x=2\]\[x\ge2\]and all these would NOT be filled-in:\[x<2\]\[x>2\]

- asnaseer

note that the first 3 INCLUDE the point x=2

- asnaseer

if that makes sense, then we can move onto the next part...

- asnaseer

?

- anonymous

yes :)

- anonymous

so stoping at two we fill in

- asnaseer

yes

- anonymous

and whatever is greater than 2 we leave alone?

- anonymous

i got it..

- asnaseer

ok, so 2nd part of the curve is defined as:\[f(x)=1\]if \(x\gt2\)
agreed?

- anonymous

yes

- asnaseer

so first lets sketch f(x) = 1

- asnaseer

|dw:1375035727853:dw|

- anonymous

wait, can't we sketch x=3 because it says x is GREATER than 2

- asnaseer

now here we are told that this part is only valid for \(x\gt2\), so we need to draw a circle (not filled-in) at the point x=2 and throw away everything to the left of this point

- asnaseer

|dw:1375035852984:dw|

- anonymous

yeah but wait, im not sure why we are using 1 when we can use 3 since it says x>2

- asnaseer

the value of f(x) = 1 for x greater than 2

- asnaseer

that includes points such as x=2.000001, x=2.5, x=3, x=99, etc

- asnaseer

i.e. ALL points to the right of (and NOT including) x=2

- asnaseer

|dw:1375035980350:dw|

- anonymous

oh okay so i can use 3.. great

- asnaseer

so now we just combine both parts of the curve to get...

- asnaseer

|dw:1375036030210:dw|

- asnaseer

make sense?

- asnaseer

|dw:1375036145535:dw|

- anonymous

got it

- asnaseer

good. now we can examine the lmits

- anonymous

ok :)

- asnaseer

the 1st one is limit as x tends to 2 from the negative side

- asnaseer

so - look at the curves and see what you think the answer should be for this case

- asnaseer

|dw:1375036264164:dw|

- anonymous

1 zt 2^-

- anonymous

at*

- anonymous

why is it -1 and not 1?

- asnaseer

?

- anonymous

i see it's coming from the left side and the answer is going to be negative.

- asnaseer

why is it going to be negative?

- anonymous

but where do i look, in the y?

- anonymous

because it's coming from the left?

- asnaseer

lets approach the limit in steps

- anonymous

how do i look for the limit when it approaches to 2?

- asnaseer

|dw:1375036393036:dw|

- asnaseer

|dw:1375036417600:dw|

- asnaseer

can you see now what the limit should be as x approaches 2 from the left-hand-side?

- anonymous

so its 3?

- asnaseer

perfect! well done! :)

- asnaseer

now try to do the same for x approaching 2 from the right-hand-side

- anonymous

right side of 2 it approaches at 1

- asnaseer

bingo! you're a pro now! :)

- anonymous

2 is undefined?

- asnaseer

the /tricky/ part now is what is the limit as x approaches 2 (without being told whether its from the left or right side)?

- asnaseer

yes!

- asnaseer

I believe you have grokked this now! :)

- anonymous

because, it's not coming from either side.

- anonymous

omg thank you!!!! :))

- anonymous

thank you so much!!

- asnaseer

yw - I'm glad I was to explain it well enough for you to understand. :)

- anonymous

i'll apply this to the rest when im doing a problem

- asnaseer

perfect! - keep at it - you seem to have a natural ability at this. :)

- anonymous

What if they tell us to calculate instead of sketching?

- anonymous

do we simply substitute?

- anonymous

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