At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
have you attempted to sketch the curve first?
which part are you stuck on?
how do i sketch if i dont know the points
all i see are x+1, x <> 2 then im stuck
i can't use calculator.
ok - lets break it down into smaller steps
first, you are told that:\[f(x)=x+1\]if \(x\le2\) agreed?
so lets sketch this...
so for a) since it;s coming from the left side it must be negative... and we use the top function, correct?
Its already been sketched btw
ok how did you sketch that ? :(
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
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
so the actual line should look like this:
does that make sense?
ohhhhh okay :)
so when are the times that we fill in the dots?
and don't fill in?
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:\[x<2\]\[x>2\]
note that the first 3 INCLUDE the point x=2
if that makes sense, then we can move onto the next part...
so stoping at two we fill in
and whatever is greater than 2 we leave alone?
i got it..
ok, so 2nd part of the curve is defined as:\[f(x)=1\]if \(x\gt2\) agreed?
so first lets sketch f(x) = 1
wait, can't we sketch x=3 because it says x is GREATER than 2
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
yeah but wait, im not sure why we are using 1 when we can use 3 since it says x>2
the value of f(x) = 1 for x greater than 2
that includes points such as x=2.000001, x=2.5, x=3, x=99, etc
i.e. ALL points to the right of (and NOT including) x=2
oh okay so i can use 3.. great
so now we just combine both parts of the curve to get...
good. now we can examine the lmits
the 1st one is limit as x tends to 2 from the negative side
so - look at the curves and see what you think the answer should be for this case
1 zt 2^-
why is it -1 and not 1?
i see it's coming from the left side and the answer is going to be negative.
why is it going to be negative?
but where do i look, in the y?
because it's coming from the left?
lets approach the limit in steps
how do i look for the limit when it approaches to 2?
can you see now what the limit should be as x approaches 2 from the left-hand-side?
so its 3?
perfect! well done! :)
now try to do the same for x approaching 2 from the right-hand-side
right side of 2 it approaches at 1
bingo! you're a pro now! :)
2 is undefined?
the /tricky/ part now is what is the limit as x approaches 2 (without being told whether its from the left or right side)?
I believe you have grokked this now! :)
because, it's not coming from either side.
omg thank you!!!! :))
thank you so much!!
yw - I'm glad I was to explain it well enough for you to understand. :)
i'll apply this to the rest when im doing a problem
perfect! - keep at it - you seem to have a natural ability at this. :)
What if they tell us to calculate instead of sketching?
do we simply substitute?