Limits and sequences

[feral current]

Hi

[bleak needle]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

+close

3. Feel free to nominate the person for helper of the week in #helper-nominations
4. Do not ping the mods, unless someone is breaking the rules.
5. If you're happy with the help you got here, and the server overall, you can contribute financially as well:

[feral current]

Assume $x_n > 0$ and $x_n \to x > 0$ as $n \to \infty \$
Show that $ \log(x_n) \to \log(x)$

[exotic minnow]

Josh

[feral current]

This is what i've done so far

The definition: $\abs{x_n - x} < \epsilon : \forall n > n_o$

[exotic minnow]

Josh

[feral current]

$0 < x - \epsilon < x_n < x + \epsilon \
\iff \log(x-\epsilon) < \log(x_n) < \log(x+\epsilon) \
\iff \log(x-\epsilon) - \log(x) < \log(x_n) - \log(x) < \log(x+\epsilon) - \log(x) \
\iff\log(\frac{x-\epsilon}{x}) < \log(x_n) - \log(x) < \log(\frac{x+\epsilon}{x}) \
\iff \log(1 - \frac{\epsilon}{x}) < \log(x_n) - \log(x) < \log(1 + \frac{\epsilon}{x})$

Can take epsilon to be arbitrary

however idk how to continue

want to somehow get something like

want to get something like $\iff -\epsilon_1 < \log(1 - \frac{\epsilon}{x}) < \log(x_n) - \log(x) < \log(1 + \frac{\epsilon}{x}) < \epsilon_1$

[exotic minnow]

Josh

Josh

[feral current]

I know that $\log(1+x) \le x$ for $x \in (-1,1) $ but can extend to something like $(-1, \infty)$

[exotic minnow]

Josh

[crimson berry]

Can you not use that if f is continuous and x_n -> x, then f(x_n) -> x?

[feral current]

-> f(x)?

[crimson berry]

My bad, yes

[feral current]

can you say that if x_n is a sequence of real numbers

[crimson berry]

Any sequence. In fact, f is continuous at x iff for all sequences x_n -> x, f(x_n) -> f(x).

But I imagine it depends on what you've proven in your course. Are you allowed to use that fact?

[feral current]

im not sure, that's the issue

"exploiting the concept of continuity"

[crimson berry]

Oh, ok

I guess you can't

[autumn wave]

I wonder who's the author. Ig that's another average guy writing another set of notes.

The result is too specific to be a theorem

It shld b sth more general like the composition of two continuous functions

[feral current]

This is first year analysis. But written by a Professor at my university

[autumn wave]

Oic that's why I immediately recall the graph of exponential function above that of y = x + 1

from which you get a lower bound for log(x)

The upper bound is easy.
The graphical way for thinking would give you a quick answer.
It's still this elementary inequality about $e^x$ and $x+1$, but

[exotic minnow]

vin100

[autumn wave]

Think: how to get a log curve from the exponential curves?
Hint: think about the graphical meaning of "inverse function"
Answer: ||reflect along the line y = x||

What do you observe? Apply this to the RHS of log(1 + ε / x), and you'll get part of your desired conclusion.

Try to describe the remaining problem verbally. Now we have the upper for the difference, and we want a lower bound. The upper bound of the target difference comes from the basic lower bound for the exponential function, so it's intuitive to interchange "upper" with "lower", and see what we can get

That's to find an upper bound in rational function for the exponential function.

You might be puzzled cuz the basic inequality $e^x \ge x + 1$ has only one direction

[exotic minnow]

vin100

[autumn wave]

I'll let you think about that cuz I gtg 🍽️

[crimson berry]

Since you can't use the fact that log is continuous, how is log defined in your class?

[worn furnace]

use the fact that sqrt(x) is continous over [0,+infty)

[crimson berry]

That's not the problem. It's trying to show that same idea but for log. I suggested using the fact that log is continuous, but OP cited that to show we can't use continuity

[worn furnace]

Oh sorry, I should've back read the conversation 😅

[feral current]

only recently covered

[autumn wave]

you don't need that

[feral current]

I didnt think so

was just responding to him

[autumn wave]

ic

it's still $e^x \ge x - 1$

[exotic minnow]

vin100

[autumn wave]

but we replaced $x$ with $-x$

[exotic minnow]

vin100

[autumn wave]

this is legitimate since this basic ineq holds for all $\bR$

[exotic minnow]

vin100

[autumn wave]

,,e^{-x} \ge -x-1

[feral current]

isnt this another question

[exotic minnow]

vin100

[autumn wave]

u might wanna see my prev msg here: #1174382035259887667 message

abt switching from symbolic to verbal way of thinking

take reciprocal on both sides

[feral current]

my mate mentioned $\abs{\log(1+\frac{\epsilon}{x}) - 0} < \epsilon_1$ for some arbitrary something

[exotic minnow]

Josh

[crimson berry]

It all depends on how things are defined and what the course allows you to assume. You can define log in terms of the integral of 1/x and then use that to define exp, for example

[feral current]

using def of limit

[autumn wave]

but here OP has explained that he's taking a real analysis course. i dun think his prof has talked abt integrals

[feral current]

$-\epsilon_1 < \log(1 - \frac{\epsilon}{x}) < \log(x_n) - \log(x) < \log(1 + \frac{\epsilon}{x}) < \epsilon_1 \
\implies -\epsilon_1 < \log(x_n) - \log(x) < \epsilon_1 \
\implies \abs{\log(x_n) - \log(x)} < \epsilon_1 \$
which by definition means $\log(x_n) \to \log(x)$

[exotic minnow]

Josh

[autumn wave]

oops typo

,,e^x \ge x + 1

[exotic minnow]

vin100

[feral current]

so $\log(1+\frac{\epsilon}{x}) \le \frac{\epsilon}{x}$

[exotic minnow]

Josh

[feral current]

from what we did before

[autumn wave]

yeah that's the upper boiund

bound*

now you need a lower bound

[feral current]

for $\log(1-\frac{\epsilon}{x})$

[exotic minnow]

Josh

[autumn wave]

that's why i'm explaining how to use your verbal intuition to get an inspiration

#1174382035259887667 message

[feral current]

from a lemma in the notes, e^x has upper bound 1/1-x

[autumn wave]

that's it

[feral current]

oh

ic

so $e^x \le \frac{1}{1-x}\
\implies x \le -\log(1-x)$ ?

[exotic minnow]

Josh

[autumn wave]

when u write a variable, form a habit: what's its quantifier and its domain?

[feral current]

lemma says for x < 1

[autumn wave]

you might wanna get idea from the graph

https://www.desmos.com/calculator/qhly36d9gu

[feral current]

i see it

but when you multiply both sides by -1, itll flip the sign

[autumn wave]

that's why you're getting lost with symbolic

remember the three ways of thinking?

symbolic, visual, and verbal

when you get lost, you might wanna switch to another sense
to get a different view, and get an inspiration

see how well the graphical approach goes

ofc u gotta justify the conclusion from the graphical approach with the symbols

[feral current]

sorry, but what am I trying to infer

[autumn wave]

you got stuck cuz what you're doing in symbols

[autumn wave]

doesn't match ur graphical observations

compare

,align
f(x) &\ge g(x) \
x &\ge f^{-1} \circ g(x)

[exotic minnow]

vin100

[autumn wave]

with

,align
f(x) &\ge g(x) \
y &\ge g(f^{-1}(y)) \tag{$y = f(x)$} \
x &\ge g \circ f^{-1} (x) \tag{$y$ replaced with $x$}

[exotic minnow]

vin100

[autumn wave]

see the difference?