Grace and L'Hôpital's Rule

Infinity

In my first-level university calculus course, infinity was difficult concept to wrap my head around. What happens when you take something infinitely large and divide it into an infinite number of pieces? What if you have an unstoppable object (infinite momentum) that collides with an immovable object (infinite inertia)? These paradoxes seem impossible to reconcile, but math tells us that not all infinities are equal. For instance, take the transcendental expression as x approaches infinity: 

Taken separately, both of the numerator and denominator clearly both reach infinity as x approaches infinity. So, what is infinity divided by infinity? At face value, infinity over infinity is indeterminate; one can't know the solution without first knowing the rate of change of each function. This is where the French mathematician Guillaume de l'Hôpital derived his famous rule: simply take the derivative of the numerator and the denominator, and the answer becomes clear. 

This makes intuitive sense. If you plug a large number, like 99, into the original transcendental equation and you get a number on the order of 10 to the power of 40. So, of course the solution would be infinity. But there would have been no way to conclude this had we not known the original expression, because infinity divided by infinity is indeterminate. l'Hôpital demonstrates that you have to know the rates of change to determine these kinds of problems. 

My professor leveraged this rule into a spiritual lesson that I will never forget. One of the joys of attending a church-owned, Christian university was that professors were always eager to share things of a spiritual nature. After explaining the rule, my professor Dr. Matthew Lewis paused and said that this rule is "a good Sunday school lesson", implicitly waiting for somebody to ask why. So, I raised my hand, and he proceeded to explain that all of us sin everyday, and thus our sins are effectively growing without bound (the definition of infinity in a calculus course). Some of us also feel that the magnitude of many of our mistakes are immensely significant, to the point that they define us and we are forever shackled by them. Combining these two effects, our sins can justifiably be modeled as infinite. 

He then pointed to the same limit expression on the board that I've provided here. "Think of the 2x as your sins", he said, "and think of e to the power of x as Christ's grace. We read in the scriptures that His atonement is infinite, but what if our sins are infinite in number, infinite in magnitude, or any combination thereof? We apply l'Hôpital's rule".  He then pointed to the derivative of the expression, and drew this on the board, which I will never forget:

"Even if our sins were infinite, they don't matter. They don't even make a dent. Christ purifies".

I had been told my whole life that Christ cleanses sins, or Christ can heal that which is broken and cannot be fixed. But to see this applied mathematically created a huge paradigm shift for me; it really doesn't matter what I do, I can't escape the infinite reach of grace. Just as Elder Jeffrey R. Holland stated, "It is not possible for you to sink lower than the infinite light of Christ’s Atonement shines". So there is really no reason to worry about dumb things in the past — Christ allows us to move past them and focus on the future.