I say that mathematicians use the words infinity and infinite, but in their computations they change the meanings of infinity and infinite, so that they are no longer with the meanings of without end, but with end, and it is because they cannot manipulate something whatever it is that is literally without end i.e. without ‘finis’.
Am I correct?
Actually, you are not correct. Mathematicians have actually developed several ways of approaching the infinite. The nice thing about having a mind is that we can approach problems that seem impossible, and do amazing things with them.
As an example, let's look at the ratio: $\frac{x^2 + x + 1000}{3x^2 - 3x + 200}$. Let's say we want to know what this looks like at $x = \text{infinity}$. Well, we can see that as $x$ grows, the importance of the term with the highest exponent ($x^2$) becomes more and more important. likewise, as $x$ gets larger, the importance of the other terms get less and less important. After $x$ gets infinitely larger, the other terms get infinitely less important, or, not important at all. Therefore, at $x = \text{infinity}$ then fraction then reduces to $\frac{x^2}{3x^2} = \frac{1}{3}$. So, even though we can't actually count to infinity, we can use the properties of infinity to reason about things in the infinite.
Now, there are multiple different conceptions of infinity, so, to deal with infinity rigorously, you have to state which conception of infinity you are using. Cantor's "cardinal numbers" deal with absolute set sizes, and shows that there are infinities of different absolute sizes. Cantor's "ordinal numbers" deal with the ordering of different ways of expressing infinite sets. My favorite way of dealing with infinity is the "hyperreal numbers", which basically give a unit value for a "base infinity", and then you can treat infinity as an algebraic unit.
For an example of how to use infinitely practically, think about the infinitely small. If we take $\epsilon$ to be an infinitely small value, we can do some interesting things with it. Let us say we want to know the value of $\frac{x^2 - 25}{x - 5}$ at $x = 5$. If you plug it in, you get $\frac{0}{0}$. However, if you nudge it slightly, you get an actual number. What if we nudged it an infinitely small amount, by $\epsilon$, so that $x = 5 + \epsilon$?. This yields: $$\frac{(5 + \epsilon)^2 - 25}{5 + \epsilon - 5} = \frac{5^2 + 2\cdot 5\cdot\epsilon + \epsilon^2 - 25}{\epsilon} = \frac{10\epsilon + \epsilon^2}{\epsilon} = 10 + \epsilon$$
In other words, if $x$ is infinitely close to $5$, the result of the expression is infinitely close to $10$.
As you can see, infinite can be used in a number of ways by mathematicians. We can do infinite sums of infinitely small numbers. We can express a calculation as a polynomial with an infinite number of terms. We can operate on series that diverge to infinity.
The key part is recognizing what infinity means in a particular context, and how the rules of that sort of infinity affect the problem you are working on.