Evaluating limits of logarithmic functions for pH modeling

[Bipolarity]

I made up this limit problems trying to understand something in chemistry, so they may not necessarily have any answer.

I am not sure how to evaluate them analytically. Not sure how L'Hopital's Rule would work here either.

$$(\lim_{V→50^{-}}-log(\frac{50-V}{10(V+50)}))$$

$$14+(\lim_{V→50^{+}}log(\frac{V-50}{10(V+50)}))$$

$$14+(\lim_{V→∞}log(\frac{V-50}{10(V+50)}))$$

I would greatly appreciate all your help on these questions!

Just in case anyone asks, I am trying to model the pH of a solution as a function of the volume of titrant added, but I want to make sure it is piece-wise continuous.

Thanks!

BiP

 

[HallsofIvy]

log(x) is continuous for x> 0 so $\lim_{V\to 50^-} log((50- V)/(10(V+ 50))= log(\lim_{V\to 50^-} (50-V)/(10(50+ V))$

And since the denominator does not go to 0 as V goes to 50, you cannot use "L'Hopital's rule" nor do you need to. As V goes to 50, from below, the numerator goes to 0 while the denominator goes to 1000 so the argument of log goes to 0 and so the limit is $-\infty$.