Many times I have read or heard that we can tell that a value approaches infinity. Yet, if infinity is not an exact value, but a general idea, how can it ever be approached? Any number that you think approaches infinity can just have one added to it, be multiplied by 7.4, or be raised to the power of itself.
Again, how can we ever say a number approaches infinity when it can just be incremented to be a little bit larger?
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Normally, if we say that a number "approaches infinity" we mean that it gets arbitrarily large. We are also normally looking at some associated phenomenon, like a limit, and in the spirit of the last sentence of your question, if the number is incremented to be "a little bit larger" we are closer to the limit than we were before.
The word "approaches" is sometimes used, but more often "tends to" which means "goes towards". These phrases are used because that means we can use the same language of infinity as we use in the finite case, and that avoids adding extra cases or putting caveats.
If you are unhappy about the language, which is conventional, then by all means use more accurate terms yourself. But do also bear in mind how other people use the words, which are not usually ambiguous.
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Again, how can we ever say a number approaches infinity when it can just be incremented to be a little bit larger?
That's the math way to say that numbers can "approach" limits that they can't necessarily reach.
It's not just about infinities, either. Consider the sequence $x_n=\frac{1}{n}$ which approaches $0$ without ever reaching it. If you find that intuitive, then so should be saying that $\frac{1}{x_n}$ approaches infinity.
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$\infty$ is not special in this regard; you could ask the same questions about approaching zero, or approaching one. The only reason $\infty$ gets special attention here is because everybody knows what zero and one are, but many don't know what $\infty$ is. (also, there is a lot of mysticism around the notion of the infinite that can make it hard for people to learn)
Also, $\infty$ is an exact value, although it is not a real number (we call it an extended real number). Do not make the mistake of confusing it with other notions like "an unspecified very large number" or "a quantity that grows without bound" or somesuch. Again, it's not really any different from numbers like $0$ and $1$.
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Many times I have read or heard that we can tell that a value approaches infinity.
$\newcommand{\Reals}{\mathbf{R}}$The phrase "approaches infinity" is convenient, compelling shorthand once careful definitions are in place, but is a recipe for trouble (confusion, logical error, ...) if taken as the final word.
To say "a number approaches infinity" is as mathematically nonsensical as saying "$2$ approaches infinity". (As the joke goes, only for large $2$.)
The mathematical objects that "approach infinity" in a meaningful sense are infinite sets, such as:
The set of natural numbers, the set of primes, the set of positive real numbers;
Functions whose domain is an infinite set, such as the sequence of natural numbers or primes in increasing order, the function $f(x) = x$ with $x$ real (which "approaches infinity as $x \to \infty$"), the function $f(x) = 1/x$ defined for positive real $x$ (which "approaches infinity as $x \to 0^{+}$"), and so forth.
Here are a few conventional definitions of "approaching infinity". These definitions are by no means general or all-encompassing; to the contrary, they're chosen for concreteness and simplicity.
Definition 1: A set $X \subset \Reals$ approaches infinity (or, more conventionally, is unbounded above) if:
For every real number $R$, there exists a number $x$ in $X$ such that $x > R$.
Definition 2a: Let $f$ be a real-valued function defined in some non-empty open interval $(a, b)$ of real numbers. The value $f(x)$ approaches infinity as $x \to b^{-}$ if:
For every real number $R$, there exists a real number $\delta > 0$ such that if $a \leq b - \delta < x < b$, then $f(x) > R$.
Definition 2b: Let $f$ be a real-valued function defined in some non-empty set $X$ of real numbers that is unbounded above. The value $f(x)$ approaches infinity as $x \to \infty$ if:
For every real number $R$, there exists a real number $M$ such that for every $x$ in $X$ satisfying $x > M$, we have $f(x) > R$.
Again, the mathematical object that approaches infinity is not a single number or function value, but an infinite set of numbers or function values. The word "approaches" re-casts a fixed, spatial certainty (the set $X$ or the set of values of $f$ exists "all at once") with a temporal potentiality of the type "If you choose a number as large as you like, I can choose a larger number lying in the set $X$ (Definition 1), or a larger function value (Definitions 2)."
Unfortunately, the term "approaches" and the breezy ways the term gets used by calculus books and teachers ("$x$ approaches $b$ but never reaches $b$", etc.) suggests that potentialities don't exist until we consider them, or that conditions can never be achieved, which leads to apprearances of infinite logical regress and expository double-talk.
Note that while it's tempting to think that mathematics is only used to model our physical reality, this is not true.
If this was true, then what sense does the number $10^{100}$ make? It's larger than the number of particles in the visible universe, so surely we can't represent it physically.
And yet, even the ancient Greek believed that if $n$ is an integer, then $n+1$ exists. So if $10^{100}$ doesn't exist, but for every $n$ which exists, $n+1$ does exist... something goes wrong.
Infinity is inherent into the natural numbers as we are used to thinking about them. For example, sets were created to allow collections of mathematical objects (like numbers) to be mathematical objects on their own accord. So naturally, we are inclined to talk about the set of natural numbers which is infinite.
Some people do reject this approach to mathematics, they may believe that infinite sets do not exist, but there are infinitely many natural numbers nonetheless; or sometimes that there is a largest number (even though we don't know what it is). These philosophical (and mathematical) schools of thought are joined under the term "finitism" (and ultrafinitism in the latter case).
Some threads of interest.