Thread: Exponential Functions: Finding The Limit

1. Exponential Functions: Finding The Limit

Find the limit.

$
\mathop {\lim }\limits_{x \to 7^ - } e^{3/(7 - x)}
$

So I said, let

$
t = 7 - x
$

If $
x \to 7^ -
$
then $
t \to 0
$

.....does $
t \to 0
$
approach from the right of left?

Sorry, i'm a bit rusty on limits. I haven't seen them in a while! Thanks for the help.

2. Hello,

Originally Posted by RedBarchetta
Find the limit.

$
\mathop {\lim }\limits_{x \to 7^ - } e^{3/(7 - x)}
$

So I said, let

$
t = 7 - x
$

If $
x \to 7^ -
$
then $
t \to 0
$

.....does $
t \to 0
$
approach from the right of left?

Sorry, i'm a bit rusty on limits. I haven't seen them in a while! Thanks for the help.
It's ok for the beginning

There's just a problem for the limit of t.

When x tends to 7-, this means that x tends to 7, and x is <7.

So t=7-x indeed tends to 0, but is t<0 or >0 ?

--------> x<7 --> -x>-7 ---> 7-x>0

Therefore, t tends to 0, with t>0, which is written " $t \to 0+$"

Is it clear for you ?

3. Originally Posted by RedBarchetta
Find the limit.

$
\mathop {\lim }\limits_{x \to 7^ - } e^{3/(7 - x)}
$

So I said, let

$
t = 7 - x
$

If $
x \to 7^ -
$
then $
t \to 0
$

.....does $
t \to 0
$
approach from the right of left?

Sorry, i'm a bit rusty on limits. I haven't seen them in a while! Thanks for the help.
To expand upon Moo's explanation

$\lim_{x\to{7^{-}}}e^{\frac{3}{7-x}}$

Means as you put a number an ifinitisemally bit smaller than 7 in you get out an infinitesimally small positive number

and a constant divided by an infinitesmially small positive number is ∞...So therefore

this is limit is not formally but informally equivalent to $e^{\infty}=\infty$

4. Originally Posted by Mathstud28
To expand upon Moo's explanation

$\lim_{x\to{7^{-}}}e^{\frac{3}{7-x}}$

Means as you put a number an ifinitisemally bit smaller than 7 in you get out an infinitesimally small positive number

and a constant divided by an infinitesmially small positive number is ∞...So therefore

this is limit is not formally but informally equivalent to $e^{\infty}=\infty$
Unless your name is Newton, Liebniz, Euler or Robinson it does not mean that at all. That is unless you know rather more than you do (and certainly more than the original poster does) about number systems there are no infinitesimals.

RonL

5. Originally Posted by CaptainBlack
Unless your name is Newton, Liebniz, Euler or Robinson it does not mean that at all. Unless you know rather more than you do about number systems there are no infinitesimals.

RonL
Ok this is a good time to ask a serious question. This in no way despite if it does is a sarcastic or mean-spirited question?

Ok on here I use non-mathematically correct terms to help explain concepts to people....should I not do that and have them possibly get bogged down in mathematical language or should I say this is a good way of conceptualizing it with the caveat that it is technically incorrect as I did above?

6. Originally Posted by Mathstud28
Ok this is a good time to ask a serious question. This in no way despite if it does is a sarcastic or mean-spirited question?

Ok on here I use non-mathematically correct terms to help explain concepts to people....
As far as I know, "infinitesimal" is a mathematical term, and Newton and Leibniz indeed have to do with this concept...

should I not do that and have them possibly get bogged down in mathematical language or should I say this is a good way of conceptualizing it with the caveat that it is technically incorrect as I did above?
Well, explaining to someone that if "a constant is divided by an infinitesimal thing, it is ∞".
∞ is just a writing convention, it's not a number.
You didn't even mention "infinity".
Not that I'm able to explain, but I try not to get into concepts I don't really master...

However, if I had to explain... :
$\frac{3}{t}$ is positive, because we showed that t > 0, and so is 3.
When you divide by a very very small number, it's like counting how many times you have this small number in the upper number (3). For example, there are 10 times 0.1 in 1. If we decrease 0.1 into 0.01, there will be 100 of these.
Here, t becomes very small, so small that there would be a very large number of them in 3...
This is the infinity...

(this latter part has nothing to do with a real explanation... i'm just trying to clarify out what has been said before)

7. Originally Posted by Mathstud28
Ok this is a good time to ask a serious question. This in no way despite if it does is a sarcastic or mean-spirited question?

Ok on here I use non-mathematically correct terms to help explain concepts to people....should I not do that and have them possibly get bogged down in mathematical language or should I say this is a good way of conceptualizing it with the caveat that it is technically incorrect as I did above?
The original question asked about a limit as x approached 7 from below. This is indicative of a non-informal approach to limits. As such use of infinitesimals are to be deprecated as an explanatory aid.

Also Bishop Berkeley's criticisms of the use of infinitesimals are relevant.

RonL