How can I say "$T \to \infty$ implies ..."?

Question

I would like to know a better way to say "$T \rightarrow \infty$ implies ..." when writing / speaking.

I tend to write this as "Limiting $T$ to $\infty$ implies ...", but others have told me that this particular phrasing sounds awkward.

Could anyone help me?

Answer 1

There are a number of different ways to phrase it. Some examples include:

• As T tends to infinity, it follows that . . .
• Taking the limit as T approaches infinity, gives the result . . .
• . . . when we take the limit as T approaches infinity
• If we limit to T to infinity, this tells us that . . .

If you are writing this mathematically rather than speaking it aloud, there are some more concise ways of putting this depending on the context. For example:

• $\lim_{x \to \infty}$ [result] = [outcome]
• As we let T $\rightarrow\infty$, [result] $\implies$ [outcome] is true
• [expression] $\rightarrow$ [value]

However, these don’t mean the same thing and so it’s important that you don’t use them interchangeably, as they are all appropriate in different contexts.

Edit: Thanks to Snaw for the suggesting the third example in the comments.