Understanding limit of exponents

[andyrk]

This might be a pretty stupid question. But why is it that while applying limits to an exponential function like- $\lim_{x\rightarrow 0} e^{f(x)}$ we move the limit to only the part of the expression which involves the variable on which the limit is being evaluated and hence we now write it as- $e^{(\lim_{x\rightarrow 0} f(x))}$? Can't we evaluate the limit without reducing the terms inside the limit?

What I mean is earlier, the limit included $e$ withing itself too. But when we simplified it, the $e$ came out of the limit and instead the limit was being operated on only $f(x)$. Can we do this? Is there any formal rule or formula for this like other formulas for evaluating and simplifying limits?

So my question is basically that is this a rule or formula which we follow or do we do it just because of logic? (The logic of applying the limit only to the part of the expression which gets affected by the limit)

 

[jedishrfu]

I think its the logic of it. Basically you could look at the expression as a mapping of x to e^f(x) and so then you could look at the x to f(x) as x approaches 0.

If you think about it knowing what happens to f(x) in the limit i.e. if it has a value means you can plug it into the e^(limit value) to get the limit value of the original expression.

 

[andyrk]

and so then you could look at the x to f(x) as x approaches 0.

What did you exactly mean by this statement? I didn't understand you fully. Earlier we were looking at x maps to e^f(x). Then how can we look at x maps to f(x) afterwards? Shouldn't it be x maps to e^f(x) all throughout?

 

[DarthMatter]

You can do this if $g(x)$ is an continuous function.

Let g(x) be a continuous function such that $\lim_{x\rightarrow x_0}g(x)=a$. Then for each $\epsilon > 0$ there is a ball $B_\epsilon$ around $x_0$ such that $|g(x)-a| < \epsilon$ for all $x \in B_\epsilon$.
Also for all $x$ $|\exp(g(x))-\exp(a)|=|\exp(g(x)-a)-1|\cdot \exp(+a)$. Now for each $\epsilon' > 0$ choose $B_{\epsilon'}$ such that $|g(x)-a| < \ln(1+\frac{\epsilon'}{\exp(a)})$. Using this in the expression above and the positive derivative of the e-function yields $|\exp(g(x))-\exp(a)| < \epsilon'$. It follows that $\lim_{x\rightarrow x_0}\exp(g(x)) = \exp(g(a))$.

 

[DarthMatter]

How can I get the arrows under the $\lim$?

 

[jedishrfu]

What did you exactly mean by this statement? I didn't understand you fully. Earlier we were looking at x maps to e^f(x). Then how can we look at x maps to f(x) afterwards? Shouldn't it be x maps to e^f(x) all throughout?

I was thinking of the calculus situation where you y=f(g(x)) and you want to find the limit when x goes to zero then you functionally decompose it by looking at the limit for g(x) and finding that plug it into f( ) to get the final answer.

 

[Stephen Tashi]

What I mean is earlier, the limit included $e$ withing itself too. But when we simplified it, the $e$ came out of the limit and instead the limit was being operated on only $f(x)$. Can we do this? Is there any formal rule or formula for this like other formulas for evaluating and simplifying limits?

As DarthMatter said, if $h(x)$ is continuous at $x = b$ and $lim_{x \rightarrow a} f(x) = b$ you can assert that $lim_{x\rightarrow a} h(f(x)) = h ( lim_{x \rightarrow a} f(x) )$.

In you example we have the special case $a = 0$ and $h(x) = e^x$. Since $e^x$ is continuous at each $x = b$ you don't have to worry about what particular $b$ results from $lim_{x\rightarrow a} f(x)$ , just as long as the limit exists.

So my question is basically that is this a rule or formula which we follow or do we do it just because of logic? (The logic of applying the limit only to the part of the expression which gets affected by the limit)

The above result is a theorem. (When people get in the habit of applying a theorem, they begin to think of it as a rule or formula). It requires some effort to prove it. Most calculus textbooks include this theorem. Is it in your course materials?

 

[Fredrik]

How can I get the arrows under the $\lim$?

\lim_{x\to a}f(x) produces $\lim_{x\to a}f(x)$ or $$\lim_{x\to a}f(x)$$ If you want the former to look like the latter, use \displaystyle like this: {\displaystyle\lim_{x\to a}f(x)}

 

[jedishrfu]

Thanks Stephen and Fredrik.

 

[andyrk]

The above result is a theorem. (When people get in the habit of applying a theorem, they begin to think of it as a rule or formula). It requires some effort to prove it. Most calculus textbooks include this theorem. Is it in your course materials?

No, I don't think it is. What is this theorem called exactly? Can you give me some URL link where I can study it more deeply?

As DarthMatter said, if h(x) h(x) is continuous at x=b x = b and limx→af(x)=b lim_{x \rightarrow a} f(x) = b you can assert that limx→ah(f(x))=h(limx→af(x)) lim_{x\rightarrow a} h(f(x)) = h ( lim_{x \rightarrow a} f(x) ) ..

How would we even get to know what b is until we have evaluated the limit? And for evaluating the limit we need to apply the theorem. But for the theorem we need b. Isn't it ambiguous?

 

[jedishrfu]

This webpage has some discussion on it:

http://www.themathpage.com/acalc/limits-2.htm

 

[andyrk]

This webpage has some discussion on it:

http://www.themathpage.com/acalc/limits-2.htm

It has discussion on constant factors inside the limit, while multiplication inside the limit is occurring. It doesn't mention anything about what happens when a constant is present in the exponential form.

 

[FactChecker]

What is this theorem called exactly?

It's not so much a theorem as it is the definition of "continuous". lim_x->ae^x = e^lim_x->ax = e^a because e^x is continuous at a.

 

[jedishrfu]

I think this is an application of the chain rule:

http://en.m.wikipedia.org/wiki/Chain_rule

 

[Stephen Tashi]

No, I don't think it is. What is this theorem called exactly? Can you give me some URL link where I can study it more deeply?

You can find the theorem by searching on "limit of composite functions" , "composite limit theorem". A link to a video (that I haven't watched myself yet) is http://www.larsoncalculus.com/calc1...he-limit-of-a-composite-function/#content-top

How would we even get to know what b is until we have evaluated the limit? And for evaluating the limit we need to apply the theorem. But for the theorem we need b. Isn't it ambiguous?

The purpose of a theorem is establish a valid mathematical conclusion from certain "givens". The techniques of working problems, often don't proceed deductively. They often assume the desired conclusion exists and work backwards. The fact that a technique of working problems uses methods that aren't not valid deductively doesn't mean there is any ambiguity about mathematical theorems. It means that the problem solving technique isn't rigorous mathematical deduction.

To evaluate the limit $\lim_{x \rightarrow a} h(f(x))$ , most people would evaluate $lim_{x\rightarrow a} f(x) = b$ and then ( if they were being careful) they would ask themselves if $h(x)$ was continuous at $x = b$.

 

[DarthMatter]

You can do this if $g(x)$ is an continuous function.

Let g(x) be a continuous function such that $\lim_{x\rightarrow x_0}g(x)=a$. Then for each $\epsilon > 0$ there is a ball $B_\epsilon$ around $x_0$ such that $|g(x)-a| < \epsilon$ for all $x \in B_\epsilon$.
Also for all $x$ $|\exp(g(x))-\exp(a)|=|\exp(g(x)-a)-1|\cdot \exp(+a)$. Now for each $\epsilon' > 0$ choose $B_{\epsilon'}$ such that $|g(x)-a| < \ln(1+\frac{\epsilon'}{\exp(a)})$. Using this in the expression above and the positive derivative of the e-function yields $|\exp(g(x))-\exp(a)| < \epsilon'$. It follows that $\lim_{x\rightarrow x_0}\exp(g(x)) = \exp(g(a))$.

Sorry, of course the last equation should be ${\displaystyle \lim_{x\rightarrow x_0} e^{g(x)}=e^a}$. Thanks for the displaystyle, Fredrik! Also the $B_{\epsilon'}$ should maybe be called by another name. I hope the idea still comes through: You can make the difference between $e^a$ and $e^{g(x)}$ arbitrarily small by making $|g(x)-a|$ 'small enough'.

I think this is an application of the chain rule:

http://en.m.wikipedia.org/wiki/Chain_rule

This I do not understand.

 

[andyrk]

To evaluate the limit $\lim_{x \rightarrow a} h(f(x))$ , most people would evaluate $lim_{x\rightarrow a} f(x) = b$ and then ( if they were being careful) they would ask themselves if $h(x)$ was continuous at $x=b$.

But why would anyone do that? Who tells them to do that? The textbook gives ways to evaluate explicit limits, not implicit. Then how can one simply evaluate limit on $f(x)$ without any valid reason/grounds to do so?

And I think that you are saying to check whether $h(x)$ is continuous at $x=a$ because $h(x)$ is within a limit. Had this not been true and had $h(x)$ been outside the limit then? I mean had it been like - $h(\lim_{x \rightarrow a} f(x) )$ instead of $\lim_{x \rightarrow a} h(f(x))$ then would you say that we need not check for whether $h(x)$ is continuous at $x=b$?

I am saying this because according to the theorem you have posted, this implies that $\lim_{x \rightarrow a} h(f(x))$ = $h(\lim_{x \rightarrow a} f(x) )$.

And since $h(\lim_{x \rightarrow a} f(x) )$ doesn't require to check whether $h(x)$ is continuous at $x=b$ or not, one need not check it. But this differs to what you said.

So had it even been $\lim_{x \rightarrow a} h(f(x))$ it would have been converted to $h(\lim_{x \rightarrow a} f(x) )$ and hence we can argue that even for $\lim_{x \rightarrow a} h(f(x))$ we don't need to check whether $h(x)$ is continuous at $x=b$ or not.

And sorry to say, but I didn't understand the proof equally well either.

 

[Stephen Tashi]

I can't understand what you are saying or asking. I suggest you consider some examples.

For example, define the function $h(x)$ as follows:
if $x < 0$ then $h(x) = 0$
if $x \ge 0$ then $h(x) = 2 + x$

Define the function $g(x)$ as $g(x) = x - 5$

Consider $\lim_{x \rightarrow 5} h(g(x))$. This limit does not exist.

Consider $\lim_{x \rightarrow 0} h(g(x))$ This limit exists and is equal to zero.

 

[andyrk]

Consider$\lim_{x \rightarrow 5} h(g(x))$. This limit does not exist

Yes, you are correct on this one. But what about - $h(\lim_{x \rightarrow 5} g(x))$? Wouldn't this be equal to 2?

 

[Stephen Tashi]

By the composite limit theorem, the limit would be $h(-5) = 0$.

 

[andyrk]

By the composite limit theorem, the limit would be h(−5)=0 .

Why would $\lim_{x \rightarrow 5} g(x) = -5$ ? Shouldn't it equal 0?

 

[Stephen Tashi]

Why would $\lim_{x \rightarrow 5} g(x) = -5$ ? Shouldn't it equal 0?

$lim_{x \rightarrow 0} g(x)= \lim_{x \rightarrow 0 } (x -5) = (-5)$.

 

[andyrk]

$lim_{x \rightarrow 0} g(x)= \lim_{x \rightarrow 0 }$ .

But it is $x→5$ not $x→0$..isn't it?

 

[Stephen Tashi]

But it is $x→5$ not $x→0$..isn't it?

Oh! - my mistake. I thought you were asking about the second example $\lim_{x \rightarrow 0} h(g(x))$.

You're asking about $h( lim_{x \rightarrow 5} g(x))$. Yes, that expression is $h(0) = 2$.

 

[andyrk]

Oh! - my mistake. I thought you were asking about the second example $\lim_{x \rightarrow 0} h(g(x))$.

You're asking about $h( lim_{x \rightarrow 5} g(x))$. Yes, that expression is $h(0) = 2$.

So that is what I am asking. In the above case i.e $h( lim_{x \rightarrow 5} g(x))$ we didn't check whether $h(x)$ is continuous at $x=0$ or not. Then why did we check whether $h(x)$ is continuous at $x=0$ for $lim_{x \rightarrow 5} h(g(x))$? Because $lim_{x \rightarrow 5} h(g(x))$ and $h( lim_{x \rightarrow 5} g(x))$ are the same expressions right? So how is it possible that we can evaluate $h( lim_{x \rightarrow 5} g(x))$ and not $lim_{x \rightarrow 5} h(g(x))$?

 

[Stephen Tashi]

So that is what I am asking. In the above case i.e $h( lim_{x \rightarrow 5} g(x))$ we didn't check whether $h(x)$ is continuous at $x=0$ or not.

That case does not involve applying the composite limit theorem. Since we don't intent to apply the theorem, we don't have to check that $h(x)$ is continuous at $x = 0$. We only have to check that $h(x)$ has a defined value at $x = 0$.

For the expression $h(lim_{x\rightarrow 5} g(x))$ to exist it is only necessary that $lim_{x \rightarrow 5} g(x)$ exists and that $h(x)$ has a defined value when it is evaluted at $lim_{x \rightarrow 5} g(x)$.

Then why did we check whether $h(x)$ is continuous at $x=0$ for $lim_{x \rightarrow 5} h(g(x))$? Because $lim_{x \rightarrow 5} h(g(x))$ and $h( lim_{x \rightarrow 5} g(x))$ are the same expressions right?

No. They are not the same. One of the expressions refers to a non-existent limit. The other is a specific number.

So how is it possible that we can evaluate $h( lim_{x \rightarrow 5} g(x))$ and not $lim_{x \rightarrow 5} h(g(x))$?

You have to think about how the example illustrates this. Let $lim_{x \rightarrow a} g(x) = L$ . The expression $lim_{x \rightarrow a} h(g(x))$ is a limit that depends on what happens then the argument of the function $h()$ is near the value $L$. By contrast the expression $h( lim_{x \rightarrow a} g(x) )$ depends only on the value of $h$ when the argument is exactly equal to $L$.

 

[andyrk]

No. They are not the same. One of the expressions refers to a non-existent limit. The other is a specific number.

If they are not the same then how come are we able to go from one expression to another by applying the theorem of composite limit?

You have to think about how the example illustrates this. Let $lim_{x \rightarrow a} g(x) = L$ . The expression $lim_{x \rightarrow a} h(g(x))$ is a limit that depends on what happens then the argument of the function h()is near the value L . By contrast the expression $h( lim_{x \rightarrow a} g(x) )$ depends only on the value of h when the argument is exactly equal to $L$ .

The expression $lim_{x \rightarrow a} h(g(x))$ doesn't depend on what happens when the argument of the function $h()$ is near the value of $L$. It should not be near $L$ but be equal to $L$ instead! I am getting really confused now! :|

 

[Fredrik]

So that is what I am asking. In the above case i.e $h( lim_{x \rightarrow 5} g(x))$ we didn't check whether $h(x)$ is continuous at $x=0$ or not. Then why did we check whether $h(x)$ is continuous at $x=0$ for $lim_{x \rightarrow 5} h(g(x))$? Because $lim_{x \rightarrow 5} h(g(x))$ and $h( lim_{x \rightarrow 5} g(x))$ are the same expressions right? So how is it possible that we can evaluate $h( lim_{x \rightarrow 5} g(x))$ and not $lim_{x \rightarrow 5} h(g(x))$?

They're not the same. Since $lim_{x \to 5} g(x)=0$, we have $h( lim_{x \to 5} g(x))=h(0)=2+0=2$. We also have $g(x)<0\Leftrightarrow x<5$ and $g(x)\geq 0\Leftrightarrow x\geq 5$. This implies that for all $x<5$, we have $h(g(x))=0$, and for all $x\geq 5$, we have $h(g(x))=2+(x-5)=x-3$. These results imply that $\lim_{x\to 5-}h(g(x))=\lim_{x\to 5} 0=0$ and $\lim_{x\to 5+}h(g(x))=\lim_{x\to 5}(x-3)=5-3=2$. (The - and + in the limit mean "from the left" and "from the right" respectively).

When you encounter an expression like $f(\lim_{x\to a}g(x))$, you can't take the limit "outside" the function (i.e. rewrite the expression as $\lim_{x\to a}f(g(x))$) unless you know that $f$ is continuous at $\lim_{x\to a}g(x)$.

 

[andyrk]

When you encounter an expression like $f(\lim_{x\to a}g(x))$, you can't take the limit "outside" the function (i.e. rewrite the expression as $\lim_{x\to a}f(g(x))$ unless you know that $f$ is continuous at $\lim_{x\to a}g(x)$

Okay, we cannot rewrite the expression $f(\lim_{x\to a}g(x))$ as $\lim_{x\to a}f(g(x))$. But we can rewrite $\lim_{x\to a}f(g(x))$ as $f(\lim_{x\to a}g(x))$, right? Doesn't that simply imply that we can rewrite $f(\lim_{x\to a}g(x))$ as $\lim_{x\to a}f(g(x))$ too?

 

[Stephen Tashi]

If they are not the same then how come are we able to go from one expression to another by applying the theorem of composite limit?

The composite limit theorem says that if certain conditions are met then the two expressions are equal. In the example of $lim_{x \rightarrow 5}h(g(x))$ the conditions are not met. So you cannot conclude the two expressions are equal.

You seem to think there is a rule that $\lim_{x\rightarrow a} h(g(x))$ can always be "rewritten" as $h(\lim_{x\rightarrow a} g(x))$ regardless of what properties $h$ has. That is a false belief.

The expression $lim_{x \rightarrow a} h(g(x))$ doesn't depend on what happens when the argument of the function $h()$ is near the value of $L$. It should not be near $L$ but be equal to $L$ instead! I am getting really confused now! :|

The $\lim_{x\rightarrow a} h(g(x))$ does depend on the values of $x$ in the vicinity of $L$.
Think about $lim_{x \rightarrow L} h(x)$.

In general, you should also know that there are examples where $lim_{x \rightarrow a} f(x) = L$ and $f(a)$ is not equal to $L$.

 

[andyrk]

The composite limit theorem says that if certain conditions are met then the two expressions are equal. In the example of $lim_{x \rightarrow 5}h(g(x))$ the conditions are not met. So you cannot conclude the two expressions are equal.

You seem to think there is a rule that $\lim_{x\rightarrow a} h(g(x))$ can always be "rewritten" as $h(\lim_{x\rightarrow a} g(x))$ regardless of what properties $h$ has. That is a false belief.



The $\lim_{x\rightarrow a} h(g(x))$ does depend on the values of $x$ in the vicinity of $L$.
Think about $lim_{x \rightarrow L} h(x)$.

The conditions are (as the video states)-

" If $f$ and $g$ are functions such that-

$lim_{x \rightarrow c} g(x) = L$

and

$lim_{x \rightarrow L} f(x) = f(L)$

then, $lim_{x \rightarrow c} f(g(x)) = f(lim_{x \rightarrow c} g(x)) = f(L)$ "

So how can you say that the conditions are not met? According to what is written above, they do.

In general, you should also know that there are examples where $lim_{x \rightarrow a} f(x) = L$ and $f(a)$ is not equal to [itex] L {/itex] .

This is possible only if $f(x)$ is not continuous at $x = a$. Right? Because for a limit to exist, the function need not be continuous at the point where the limit is being evaluated. Am I right?

 

[Fredrik]

The conditions are (as the video states)-

" If $f$ and $g$ are functions such that-

$lim_{x \rightarrow c} g(x) = L$

and

$lim_{x \rightarrow L} f(x) = f(L)$

then, $lim_{x \rightarrow c} f(g(x)) = f(lim_{x \rightarrow c} g(x)) = f(L)$ "

So how can you say that the conditions are not met? According to what is written above, they do.

In the example with h(x)=0 for x<0, h(x)=2+x for x≥0 and g(x)=x-5 for all x, only the first of these conditions is met. We have $\lim_{x\to 5} g(x)=g(5)=0$, but the limit $\lim_{x\to 0} h(x)$ doesn't exist.

The second condition in the quote above can be thought of as the definition of what it means to say that f is continuous at L.

This is possible only if $f(x)$ is not continuous at $x = a$. Right? Because for a limit to exist, the function need not be continuous at the point where the limit is being evaluated. Am I right?

Yes.

 

[Fredrik]

Consider this simple example. Let $f$ be the function defined by
$$f(x)=\begin{cases}
1 & \text{if }x=0\\
0 & \text{if }x\neq 0
\end{cases}.$$ We have $f(\lim_{x\to 0} x)=f(0)=1\neq 0=\lim_{x\to 0}f(x)$. Clearly the probolem is that $f(0)$ isn't equal to $\lim_{x\to 0}f(x)$, i.e. that $f$ isn't continuous at $0$.

 

[andyrk]

In the example with h(x)=0 for x<0, h(x)=2+x for x≥0 and g(x)=x-5 for all x, only the first of these conditions is met. We have $\lim_{x\to 5} g(x)=g(5)=0$, but the limit $\lim_{x\to 0} h(x)$ doesn't exist.

The second condition in the quote above can be thought of as the definition of what it means to say that f is continuous at L.


Yes.

Could you explain me the proof for this theorem? I have it but I don't understand it well enough-

"For a given $ε > 0$, find $∂ > 0$ such that-
$| f(g(x)) - f(L) | < ε$
Whenever $0 < |x - c| < ∂$"

Now as to how to find such $ε$ such that $ε > 0$-

$| f(u) - f(L) | < ε$ Whenever | μ - L | < ∂₁ (Didn't understand this at all)

| g(x) - f(L) | < ∂₁ , whenever $0 < |x - c| < ∂$ (didn't understand it again)

Let $μ = g(x)$ , $| f(g(x)) - f(L) | < ε$, whenever $0 < |x - c| < ∂$. (didn't understand this either)

 

[andyrk]

We have $f(\lim_{x\to 0} x)=f(0)=1\neq 0=\lim_{x\to 0}f(x)$. Clearly the problem is that $f(0)$ isn't equal to $\lim_{x\to 0}f(x)$

Okay, but first you said that $f(0)$ is equal to $\lim_{x\to 0}f(x)$. Then you say that it isn't. But I think it is. Which one am I supposed to believe anyways?

And anyhow, this example didn't make me understand the proof of the theorem even though it was really good and I appreciate you for that. :)