Is \( \infty + 1 > \infty \)?
This is part of a series on common misconceptions.
Is this true or false?
\[ \infty + 1 > \infty \]
Why some people say it's true: Adding one to any number always makes it bigger.
Why some people say it's false: Infinity is not a real number, so this arithmetic is not valid. Also, according to the infinity rules, \(x + \infty = \infty\), with \(x\) is any real number. Hence, \(\infty + 1 = \infty\)
The statement is \(\color{red}{\textbf{false}}\).
Proof:
We cannot do ANY ARITHMETIC OPERATION with infinity, since it is a concept, not a number.
But let's take a look at the limit to prove it is wrong.
Let \[\lim_{x\to\infty}{f(x)} = \infty\] and \[\lim_{x\to\infty}{g(x)} = 1\]
We have: \[\lim_{x\to\infty}{f(x)} + \lim_{x\to\infty}{g(x)} = \lim_{x\to\infty}{(f(x)+g(x))} = \infty + 1 = \infty\]
Hence, the statement is \(\color{red}{\textbf{false}}\).
Rebuttal: But what if you have a very large number, say, \(x = 3141592653589\dots\), will it be larger infinity?Reply: You may want to see Is \( \infty + \infty > \infty \) ?
Rebuttal: If you cannot do any arithmetic operation with infinity, what does \(\infty + 1 = \infty\) mean in your proof?
Reply:
That means you use the one of the properties of infinity to evaluate the limit of a function.
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