Rather than just comparing the numerator and denominator directly, let’s compare their derivatives (i.e., rate of change)!Īnd this is the idea behind L’Hospital’s Rule!īy comparing the rate of change, we can easily decipher the function’s behavior as a whole. Hmm, if only there were a way to observe their individual rate of change, then we can see which function is the dominant function. Is the numerator rapidly approaching infinity while the denominator is going to infinity more slowly? Or is the denominator speeding toward infinity, and the numerator is lagging behind? In other words, who is dominating the behavior of the overall limit? Is it the numerator or denominator? This means the limit of this algebraic fraction is indeterminate, and the limit doesn’t give us a clear picture of what is happening. Suppose we are taking the limit of a function as x approaches infinity and whose numerator and denominator subsequently approach infinity as well. When I was studying calculus for the first time, this was the explanation given to me, and now I want to pass it along to you.

Okay, so without getting bogged down with semantics or the formal proof involving Cauchy’s Mean Value Theorem, let’s discuss why L’Hopital’s Rule works. Observe that we had to apply L’Hopital’s rule twice to find the limit value.īut overall, the process is straightforward: if the limit is indeterminate, take the derivative of the top and the derivative of the bottom separately and then reevaluate the limit until you arrive at a defined value.

Limit zero over zero