# Left-handed limit of a rational function

1. Feb 21, 2015

### lep11

1. The problem statement, all variables and given/known data
What is Lim (1+x2)/(4-x) as x approaches 4 from the left? Prove using the definition.

2. Relevant equations

3. The attempt at a solution
Well x≠4. Function approaches positive infinity as x approaches 4 from the left side. Let m>0 and 0<x<4.
Then (1+x2)/(4-x) > x2/(4-x) > x/(4-x) > m when x...here I am stuck.
So basically I need to show that if x>xm then (1+x2)/(4-x) >m?
But at some point the same function approaches negative infinity so do a choose xm=(something, 4)?

Last edited: Feb 21, 2015
2. Feb 21, 2015

### PeroK

It's usually a good idea to simlify things as much as possible before you start.

What can you say about $1+x^2$ and $1$?

3. Feb 21, 2015

### Svein

Well, (1 + x2)≥1 for all x. So for x<4, (1+x2)/(4-x)>1/(4-x). Now take an M>0. If (4-x)<1/M, then (1+x2)/(4-x)>M.

Last edited: Feb 21, 2015
4. Feb 21, 2015

### lep11

When to use M and m? Does it matter?

5. Feb 21, 2015

### lep11

So 4 - 1/M < x < 4 ⇒(1+x2)/(4-x) > M >0 Q.E.D Thanks for super fast replies!

Last edited: Feb 21, 2015
6. Feb 21, 2015

### HallsofIvy

As long you define M and m, it doesn't matter. In most textbook proofs, perhaps just because "M" is bigger than "m", "M" is use for a "maximum" or upper bound, "m" for a "minimum" or lower bound.