# How to interpret f(x) tends to g(x)

• I
• Gaussian97
In summary, the phrase "The function ##f(x)## tends to the function ##g(x)## when x tends to ##x_0##" is defined ambiguously in a mathematical way. It can mean either ##\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)## or ##\lim_{x\to x_0} \frac{f(x)}{g(x)}=1##, depending on the context. However, in the context of a sequence of functions or a parameterized family of functions, it is often interpreted as "pointwise convergence". This definition can also be extended to a Hilbert space inner product, where the distance
Gaussian97
Homework Helper
TL;DR Summary
How to interpret the phrase "The function f(x) tends to the function g(x) when x tends to x0"
Hi, I have a question, sometimes one sees in exercises or textbooks some phrase like
The function ##f(x)## tends to the function ##g(x)## when x tends to ##x_0##
My question is, is this unambiguously defined in a mathematical way? I mean, when one reads such a thing this could mean that
$$\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)$$
Another way (I think) this can be interpreted is that
$$\lim_{x\to x_0} \frac{f(x)}{g(x)}=1$$
But also, in addition to ##\lim_{x\to x_0} f(x)=\lim_{x\to x_0} g(x)## one can think also to include a more restrictive condition
$$\lim_{x\to x_0} f'(x)=\lim_{x\to x_0} g'(x)$$
And so on..., even you could try to define that both function must have the same Taylor series arround ##x_0## etc...

So is there a unambiguos definition of that phrase or simply something that physicist say that depend completely on the point of view of the person?

S.G. Janssens
That doesn't make much sense at all. If ##x \to x_0## then ##f(x)## tends to a single value ##f(x_0)## which is not a function.

Maybe it is meant as ##x\to \pm \infty## in which case it means that ##\lim_{x \to \pm \infty} |f(x)-g(x)|=0##.

fresh_42 said:
That doesn't make much sense at all. If ##x \to x_0## then ##f(x)## tends to a single value ##f(x_0)##
Provided ##f## is continuous at ##x_0##.

S.G. Janssens said:
Provided ##f## is continuous at ##x_0##.
Yes, but even if ##f## is not continuous, then it still converges to a point in ##\mathbb{R}\cup \{\pm \infty\}##, which is no function either.

Gaussian97 said:
My question is, is this unambiguously defined in a mathematical way?

In a manner of speaking, it is defined ambiguously in a mathematical way!

I don't know what textbook you are quoting. One given function ##g(x)## cannot "approach" a different function. There can be situations like ##lim_{x\rightarrow \infty} g(x) = lim_{x \rightarrow \infty} f(x)## but this does not say that ##g(x)## and ##f(x)## "approach" each other at all values.

The context in which functions "approach" other functions is when we are talking about a sequence of functions or parameterized family of functions. For example, you can think of the expression ## f(x,a) = e^{ax}## as defining a parameterized family of exponential functions.

We can speak of a limit of a sequence of functions being a specific function. However "the limit" of a sequence of functions is an ambiguous phrase because there are several distinct definitions for what it means for a sequence of functions to converge to another function. Examples of distinct limit concepts for sequences of functions are "pointwise convergence", "convergence in L2", "uniform convergence". These are often discussed in the context of random variables in probability theory where a limit of a sequence of random variables often amounts to finding a limit of a sequence of their probability distributions.

For a parameterized family of functions ##f(x,a)##, a text may treat ##f## as a function of two variables and speak of ##lim_{a \rightarrow a_0} f(x,a)## as defining a function. As a statement about functions, I interpret this to mean "pointwise convergence".

fresh_42 said:
Yes, but even if ##f## is not continuous, then it still converges to a point in ##\mathbb{R}\cup \{\pm \infty\}##, which is no function either.
No, this is not true. If ##f## is not continuous at ##x_0##, then ##f(x)## need not converge at all as ##x \to x_0##.

S.G. Janssens said:
No, this is not true. If ##f## is not continuous at ##x_0##, then ##f(x)## need not converge at all as ##x \to x_0##.
... in which case "tends to" isn't even defined. The question implicitly requires the existence of a limit, for otherwise it wouldn't make any sense either.

fresh_42 said:
... in which case "tends to" isn't even defined. The question implicitly requires the existence of a limit, for otherwise it wouldn't make any sense either.
This is opaque and besides the issue.

You made a mistake in post #2.
I briefly pointed it out.
There is nothing more to it.

member 587159
S.G. Janssens said:
You made a mistake in post #2.
I briefly pointed it out.
There is nothing more to it.
Yes, I have assumed a property which wasn't given. But neither of your objections makes ##f(x)## tend to ##g(x)##. Hence the core statement, that if at all, then it is a number and no function, remains valid.

I think "is asymptotic to" is more useful than "tends to" here.

This requirement comes down to $$\lim_{x \to x_0} \frac{f(x) - g(x)}{g(x)} = 0,$$ which if $f$ and $g$ are continuous is equivalent to $$\lim_{x \to x_0} \frac{f(x)}{g(x)} = 1.$$

PeroK
You have to select a norm for the function space. The obvious one (to me) is to use a Hilbert space inner product of the type $<f, g> = \int \lvert f\rvert \cdot \lvert g \rvert$ and define the distance d(f, g) as $\sqrt{<f-g, f-g>}$.

I think physicist use this language to denote what mathematicians call an "Equivalence class", that is ##f(x) \sim g(x)## as ##x \rightarrow x_0## but it's been awhile since I've thought about those structures, but that's how I view it.

It should also be noted that $\lim_{x \to x_0}(f(x) - g(x)) = 0$ can hold in circumstances where neither $\lim_{x \to x_0} f(x)$ or $\lim_{x \to x_0} g(x)$ exist separately.

For example, it would make sense to say that $x(1 - e^{-x})$ "tends to" $x$ as $x \to +\infty$.

FactChecker
Indeed, f(x)-g(x) and f(x)/g(x) can have different behaviours when at x0, either both f(x) and g(x) approach zero or both approach infinity.

## 1. What does "f(x) tends to g(x)" mean?

When we say "f(x) tends to g(x)," it means that as the input value of x gets closer and closer to a certain number, the output value of f(x) approaches the output value of g(x). In other words, the behavior of f(x) becomes more and more similar to that of g(x) as x approaches a particular value.

## 2. How do you interpret the limit of f(x) as x approaches a certain value?

To interpret the limit of f(x) as x approaches a certain value, we need to look at the behavior of f(x) as x gets closer and closer to that value. If the output values of f(x) approach a specific number as x approaches the given value, then that number is the limit of f(x) at that value.

## 3. Can the limit of f(x) exist if f(x) is undefined at the given value?

Yes, the limit of f(x) can exist even if f(x) is undefined at the given value. This can happen if the left and right limits of f(x) are equal or if the function has a removable discontinuity at that value.

## 4. What does it mean if the limit of f(x) does not exist?

If the limit of f(x) does not exist, it means that the behavior of f(x) as x approaches the given value is not consistent. This could happen if the left and right limits of f(x) are not equal or if the function has a non-removable discontinuity at that value.

## 5. How can we use the concept of "f(x) tends to g(x)" in real-world applications?

The idea of "f(x) tends to g(x)" is often used in calculus to analyze the behavior of functions and their limits. In real-world applications, this concept can be applied to situations where we want to predict the behavior of a system or process as a variable approaches a certain value. For example, we can use it to predict the growth rate of a population as the time approaches a certain value, or to estimate the speed of a moving object as it approaches a particular location.

• Calculus
Replies
22
Views
1K
• Calculus
Replies
9
Views
2K
• Calculus
Replies
16
Views
3K
• Calculus
Replies
24
Views
2K
• Calculus
Replies
3
Views
2K
• Calculus
Replies
30
Views
2K
• Calculus
Replies
1
Views
986
• Calculus
Replies
3
Views
1K
• Calculus
Replies
4
Views
1K
• Calculus
Replies
11
Views
1K