Calculus: Am doing some homework, have no idea how to approach this problem. Would appreciate if someone could take some time to walk me through steps.
#Delta and Epsilon help
devout falcon
cosmic trail
all of them or a particular part?
devout falcon
The whole thing, I've been pretty lost on this topic
cosmic trail
ok.
devout falcon
I have the solutions if needed, but I've tried working backwards and confused myself
cosmic trail
no, don't work backwards from the answer key.
devout falcon
Yeah I fried my brain 😢
cosmic trail
so given $4<x<4+\epsilon$, we have $2<\sqrt{x}<\sqrt{4+\epsilon}$. (Everything is non-negative so we're fine). Now, We need to bound the right hand side. The hint (which is pretty obvious algebra) states that $4+\epsilon<(2+\epsilon)^2$, which implies that $\sqrt{4+\epsilon}<2+\epsilon$, so by transitivity of inequality we have the result. Note that $\epsilon>0$ is crucial as it makes taking square root a legit operation, so you should constantly refer to that in a formal proof.
lyric nicheBOT
Kevin S
devout falcon
Ok ok things are starting to make a little sense
so this section is just basic algebra
if epsilon was not restricted to just being >0, we would have +- versions which would muddle up everything right>
cosmic trail
Part b) is a bit tricky. We should immediately identify that $f(x)=\sqrt{x}$ and $L=2$. Now, since we require $4<x<4+4\epsilon$, it means that we are trying to approach to $4$ from the right hand side, so the limit is $$\lim_{x\to 4^+}\sqrt{x}=2$$
lyric nicheBOT
Kevin S
cosmic trail
it's a one-sided limit.
cosmic trail
why is it not f(X) as x approaches 2 from the right?
think about the definition of limit.
devout falcon
because sqrt 4 is 2
cosmic trail
you are trying to arrive at |f(x)-L|<epsilon at the end. so clearly f(x) should be sqrt x and L=2.
Is part b ok?
devout falcon
why do we not use x instead of sqrt x
cosmic trail
can you elaborate that question?
devout falcon
you typed lim (sqrt x) as x approaches 4 = 2
cosmic trail
4^+
devout falcon
lim x as x approaches 2+ = 2?
cosmic trail
Well, if you're trying to prove $\lim_{x\to 2^+}=2$, then why do you need all those extra stuff like $4<x<4+4\epsilon$? The proof of that is just trivial, given $\epsilon>0$, there exists $\delta=\epsilon$ such that $0<x-2<\delta\implies |x-2|<\delta=\epsilon$.
lyric nicheBOT
Kevin S
devout falcon
ok thanks so much for clarifyin g
cosmic trail
so let's move on to part c.
Now, part c) is just to complete the proof. Given part a), we know that given an arbitrary $\epsilon>0$, if $4<x<4+4\epsilon$, then we have the desired implication, of $|f(x)-L|<\epsilon$. The issue is how to choose a value of delta that depends on epsilon. Well, that's just the given inequality, so take $\delta=4\epsilon$, we thus have that given $\epsilon>0$, there exists $\delta=4\epsilon>0$ such that $0<x-4<\delta$ (recall the inequality $4<x<4+4\epsilon$ in part a)) implies $|\sqrt{x}-2|<\epsilon$. This implication holds precisely because of part a).
lyric nicheBOT
Kevin S
devout falcon
what exactly is the given inequality is it 4<x<4+4epsilon?
cosmic trail
what exactly is the given inequality is it 4<x<4+4epsilon?
in part a.
that is the assumption in part a, and we need to use this inequality to choose delta so that the implication (of proving limit) is true.
devout falcon
2<sqrt(x)<2+epsilon
cosmic trail
2<sqrt(x)<2+epsilon
this is the thing we want to arrive at after choosing delta.
devout falcon
did you get to delta by subtracting 4 from 4<x<4+4epsilon?
Sorry if I'm super slow
cosmic trail
did you get to delta by subtracting 4 from 4<x<4+4epsilon?
we want to choose delta such that $0<x-4<\delta$ implies $|\sqrt{x}-2|<\epsilon$.
lyric nicheBOT
Kevin S
cosmic trail
now, this implication holds when $\delta =4\epsilon$, which is obtained precisely by manipulating $4<x<4+4\epsilon$ in part a). So you're right!
lyric nicheBOT
Kevin S
devout falcon
sorry I keep asking questions so in similar problems would we obtain delta by manipulating the function to isolate epsilon?
so if the function were like 5<x<5+5epsilon would delta in terms of epsilon be 5epsilon?
cosmic trail
sorry I keep asking questions so in similar problems would we obtain delta by ma...
We are trying to obtain $0<x-4<\text{something}$, not isolating epsilon.
lyric nicheBOT
Kevin S
devout falcon
I'm still so confused about part c
I get up to f(x) -2 <epsilon
we have to choose the value of delta based on the given inequality?
cosmic trail
we have to choose the value of delta based on the given inequality?
the formal process is to choose delta depending on epsilon so that we can have the desired implication.
that's usually hard, but in this case we have that given inequality which can help us make a decision about delta.
devout falcon
so is it kind of like guessing based on the given inequality?
cosmic trail
so is it kind of like guessing based on the given inequality?
it's not actually guessing. In general when we do a formal proof, that epsilon-delta implication is what we really want, so usually we will "cheat" a little bit by working backwards from the implication, i.e., if |f(x)-L|<epsilon is true, then what should delta (depends on epsilon) be? Part a) addresses this in your problem.
devout falcon
oh wait hold up
ohhh
ok because we found limit in part 2
and then the thing for delta is 0<x-c<delta
found that c is 4
0<x-4<delta
the inequality is 4<x<4+4e
which happens to evaluate to 0<x<4e
4e=delta
that chain of logic makes sense or did i get lucky?
cosmic trail
I think you got the idea.
altho a bit unorganized, that's not a big issue as you come across more examples/exercises in the future.
devout falcon
ok thank you so much! Sorry for any headaches given along the way
Really appreciate it, this problem has been gnawing on what's left of my brain
cosmic trail
nonono it's okay, I also spent a lot of time when I studied this concept two yrs ago.
devout falcon
college student?
cosmic trail
I will be a college student.
cosmic trail
no, i'm not. just self-study.
devout falcon
from the states?
cosmic trail
not from the US.
devout falcon
oh where you from?
cosmic trail
from this earth.
devout falcon
fair enough, have you already applied to colleges?
cosmic trail
yeah, will be a college student in a few weeks.
devout falcon
ah nice good luck and thanks for the help again man