Question

find the limit help

Answer 1

\[\lim_{x \rightarrow -2^{-}}\frac{ 2x ^{2}+x+1 }{ x+2 }\]

Answer 2

@SithsAndGiggles I need some help if you are not busy please

Answer 3

Hmm so if you plug x=-2 directly in, it tells us that this function is approaching an indeterminate form of \(\Large \frac{7}{0}\), right?

Answer 4

yes

Answer 5

That's telling us that we're approaching an asymptote.
So we're approaching positive or negative infinity.

To determine whether it's positive or negative infinity, let's look at the signs of our fraction.
The numerator is approaching a positive 7.

If we're approaching from the left side of -2, is the denominator approaching from the negative or positive side?

Example:\[\Large x+2 \qquad\to\qquad -2.1+2 =?\]

Answer 6

-.1

Answer 7

so it is negative infinity

Answer 8

So I am able to say that this is a limit that is approaching infinity/does not exist because it is a nonremovable vertical asymptote

Answer 9

mmm yah that sounds like a good way to describe it! :)

Answer 10

alright thanks

Answer 11

can I ask one more I did some work on it then got stuck

Answer 12

sure

Answer 13

\[\lim_{\Delta x \rightarrow 0} \frac{ \sin[(\pi/6)+\Delta x]-1/2 }{ \Delta x}\]
so I did the identity sin(A+B)=sinAcosB=cosAsinB and then simplified and split up the fraction to get
\[\frac{ 1/2\cos \Delta x }{ \Delta x }+\frac{ \sqrt{3} }{ 2 }-\frac{ \frac{ 1 }{ 2 } }{ \Delta x }\]

Answer 14

now I am not sure how to porgress

Answer 15

Hmm I'm confused what this is...
Was it supposed to be the derivative of sin x, using the limit definition of the derivative, evaluated at pi/6?

Answer 16

It is just finding the limit as delta x approaches zero and the hint given was \[\sin(\Theta+\phi)=\sin \Theta \cos \phi +\cos \Theta \sin \phi \]

Answer 17

and so I broke up into that and I know sinpi/6=1/2 and cos pi/6=sqrt3/2

Answer 18

Using the `Sine Angle Sum Identity` gives us,\[\Large \lim_{\Delta x\to0}\frac{\sin(\pi/6)\cos \Delta x+\cos(\pi/6)\sin \Delta x-\sin(\pi/6)}{\Delta x}\]
Ok looks good so far.
It looks like you uhh converted the sines and cosines of pi/6, let's not do that just yet.

Answer 19

We're going to do some grouping, then we can apply a couple of weird identities.

Answer 20

\[\Large \lim_{\Delta x\to0}\sin(\pi/6)\color{orangered}{\frac{\cos \Delta x-1}{\Delta x}}+\cos(\pi/6)\color{royalblue}{\frac{\sin \Delta x}{\Delta x}}\]
Take a look at that and lemme know if you're confused by the way I grouped those.
I factored a sin(pi/6) out of a couple terms.
Ignore the colors just for now :)

Answer 21

I should have put brackets around the orange part so it's a little clearer.

Answer 22

If that's too confusing, the step previous to that would have looked like this:\[\Large \lim_{\Delta x\to 0}\frac{\sin(\pi/6)\cos \Delta x-\sin(\pi/6)}{\Delta x}+\frac{\cos(\pi/6)\sin \Delta x}{\Delta x}\]

Answer 23

Comeon speak up XD Where's the confusion lie? :D

Answer 24

lye* lie? whatever :3

Answer 25

I dont understand how you got -1 everything else is good

Answer 26

i see how you put sinpi/6 but isnt that 1/2

Answer 27

Factoring sin(pi/6) out of each of the first 2 terms.\[\Large \lim_{\Delta x\to 0}\frac{\color{#CC0033}{\sin(\pi/6)\cos \Delta x-\sin(\pi/6)}}{\Delta x}+\frac{\cos(\pi/6)\sin \Delta x}{\Delta x}\]Becomes:\[\Large \lim_{\Delta x\to 0}\frac{\color{#CC0033}{\sin(\pi/6)(\cos \Delta x-1)}}{\Delta x}+\frac{\cos(\pi/6)\sin \Delta x}{\Delta x}\]

Answer 28

Ya I don't wanna simplify that yet though, it's going to get sloppy if we do D: It's prettier this way.

Answer 29

I see you are factoring out a sin pi/6 but why would that get sloppy couldn't you just factor out a 1/2 if it was simplified?

Answer 30

I mean it's fine if you'd rather do it that way.
The problem was: Find the derivative of sin x using the limit definition, evaluated at pi/6.

I don't know if you've learned trig derivatives at this point, but it's leading to an important derivative.

Answer 31

no we have not

Answer 32

thank you

Answer 33

Sigh, this sucks :( I'm going to have to switch over to FireFox for OpenStudy I think. Chrome can't handle the LaTeX Plugin anymore T.T
Browser keeps crashing :c grr

Answer 34

Fine fine fine we can do it the sloppy way if you want :) lol
So right now we have this, yes?
\[\Large \lim_{\Delta x\to0}\frac{1}{2}\cdot\color{orangered}{\frac{\cos \Delta x-1}{\Delta x}}+\frac{\sqrt{3}}{2}\cdot\color{royalblue}{\frac{\sin \Delta x}{\Delta x}}\]

Answer 35

? +_+