Length of hypotenuse v/s change in height of the opposite

Imagine a right-angle triangle with opposite side of $x$, that you'll keep changing. An adjacent side $a$ a constant that you'll leave alone and fix. Then the length of the hypotenuse is given by $$\sqrt{x^2 + a}$$

Now, as $x$ grows very large, the constant $a$ becomes kind of irrelevant. Think of adding $10$ to $20,000,000,000,000$ it'll hardly make a difference to the calculuations, so we can say that $$\sqrt{x^2 + a} \approx \sqrt{x^2} = x$$ for large $x$. For the smaller values of $x$, that extra $+a$ will have quite an impact on the size of the $x^2 + a$ term.

Let me try a slightly different intuition.

Suppose you are out in the open somewhere and you see a hungry lion $60$ meters west of you and $10$ meters south. The lion runs faster than you, so your only hope for survival is to put as much distance between you and the lion as quickly as possible before it starts to chase you, and hope that a helicopter will arrive and rescue you before the lion catches you.

(If you prefer, substitute "velociraptor" or "zombie" everywhere you see "lion" in this answer.)

At the start of this scenario, you and the lion are at vertices of a right triangle whose $60$-meter leg is adjacent to the lion and whose $10$-meter leg is adjacent to you:

Now it should be fairly evident from the picture that while running due north will put some extra distance between you and the lion, it does not increase the distance nearly as quickly as running in the direction labeled "best escape route" (in the same direction as the hypotenuse of the triangle, slightly north of east).

Your calculations should bear out the fact that the distance to the lion (the length of the hypotenuse) does not grow nearly as fast as the distance you have run.

But suppose you were already $600$ meters north of the lion, not $10$, at the moment you see each other. This is the situation in the diagram below:

The "best escape route," which will maximize the distance you put between you and the lion, is still in the direction of the hypotenuse, but now that direction is nearly (though not quite) due north. So if you happen to run due north, you will not get quite as far from the lion as you might, but you will get very nearly as far as possible as quickly as possible.

The "run due north" strategy gets better and better the farther north you start, because the difference between due north and "directly away from the lion" gets less and less. But you can never get away faster than running directly away, so that is the limit on how fast the "run due north" strategy can put distance between you and the lion.

In other words, the rate of change of the hypotenuse starts small, increases as you increase the length of the leg, but eventually starts to approach a maximum value. Therefore it cannot increase linearly.