"Most people seem to struggle with this fact when first introduced to calculus..."
When I took calculus in college, I _did not_ struggle with this idea, despite at that time not having had any deeper background in advanced mathematics. Intuitively, the idea that you _approach_ some absolute as you edge the denominator ever larger made perfect sense to me.
Formally, my instructor made it clear that the _limit_ as you approach something was, in nature, different from any particular fixed value (of x). So, in a clearly defined manner, as you _apply the limit operator_ to the left side of the equation, the right side correspondingly behaves differently.
This concept never troubled me. As other comments here imply, this is an idiom specific to (differential) calculus. The only caveat might be in the use of strict equality, since limit operations by definition indicate asymptotic behavior. One could argue that a different type of relation is described (such as 'approximately equal': ≈). But then it's not infinity itself which is at issue.
When I took calculus in college, I _did not_ struggle with this idea, despite at that time not having had any deeper background in advanced mathematics. Intuitively, the idea that you _approach_ some absolute as you edge the denominator ever larger made perfect sense to me.
Formally, my instructor made it clear that the _limit_ as you approach something was, in nature, different from any particular fixed value (of x). So, in a clearly defined manner, as you _apply the limit operator_ to the left side of the equation, the right side correspondingly behaves differently.
This concept never troubled me. As other comments here imply, this is an idiom specific to (differential) calculus. The only caveat might be in the use of strict equality, since limit operations by definition indicate asymptotic behavior. One could argue that a different type of relation is described (such as 'approximately equal': ≈). But then it's not infinity itself which is at issue.