did i use infinite wrong?

Question

This algebra question is in Dutch and the original file van be found here: Question 19

Ill try to translate the important info needed to answer this question.

$$s= \frac{(a+b)} { (ab)}$$

S= dpt

a= distance(in meters) between eye lens and object

b= distance (in meters) between eye lens and retina

additional info: for his left eye $b=0,017$ meters and he can see objects sharp from a distance (a) of $15\times10^{-2}$ meters and further.

question 19: Between which 2 values of S can this person see/ wear on his glases ?

My answer so far:

$$s= \frac{15\times10^{-2} + 0,017} {15\times10^2 \times 0,017}$$

$$s= 65ish$$

My question:

How do I calculate the other value ? if I fill infinite in the formula I come up with 1 which isn't good. $\frac{big value}{big value} = 1$ right, or am i not understanding correctly how to use infinite in these situations? if so how is the correct way to use infinite?

Answer according to answer sheet: answer sheet

Answer 1

Hmm, after trying a squinty-eyes interpretation of the Dutch, it looks like the question is asking for both the maximal and minimal value that $\frac{a+b}{ab}$ can take for any $a\in[0.15,\infty)$ when $b=0.017$.

In other words you're interested in the behavior of $$ \frac{a+0.017}{a\cdot 0.017}$$ for large $a$. We can rewrite this as $$ \frac{a+0.017}{a\cdot 0.017} = \frac{a}{a\cdot 0.017} + \frac{0.017}{a\cdot 0.017} = \frac{1}{0.017} + \frac{1}{a} $$ We can see that this is a decreasing function of $a$ -- and when $a$ is large the value is close to, but slightly larger than, $\frac{1}{0.017}$. It can become as close to $\frac{1}{0.017}$ as you want by choosing a large $a$.

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You can't just say "large value divided by large value" and expect the result to be $1$. For example, a billion and two billion are both large numbers, but their quotient is $\frac12$, not $1$.

Answer 2

Hmmm....

Couldn't we say $\frac {a+b}{ab} = \frac a {ab} +\frac b {ab}= \frac 1b +\frac 1a $? Thus if $a $ is "infinite" then $\frac {infinite + b}{infinite * b} =\frac {infinite}{infinite*b} =\frac 1b$ is consistent with $\frac 1b + \frac 1 {infinite}= \frac 1b +0 =\frac 1b $.

I think your confusion is inf/inf can be anything-- not just 1. In the case of inf/inf x b, it should be 1/b. Assuming the infs are "the same".

In actuality you can't really do any of this. But you can do $\lim_{n\rightarrow \infty}\frac {n +b}{nb} = \lim 1/b + 1/n = 1/b $.