Left-handed limit of a rational function

[lep11]

Homework Statement

What is Lim (1+x²)/(4-x) as x approaches 4 from the left? Prove using the definition.

Homework Equations

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>x_m then (1+x²)/(4-x) >m?
But at some point the same function approaches negative infinity so do a choose x_m=(something, 4)?

 

[PeroK]

Homework Statement

What is Lim (1+x²)/(4-x) as x approaches 4 from the left? Prove using the definition.

Homework Equations

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>x_m then (1+x²)/(4-x) >m?
But at some point the same function approaches negative infinity so do a choose x_m=(something, 4)

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 $1$?

 

[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.

 

[lep11]

When to use M and m? Does it matter?

 

[lep11]

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

 

[HallsofIvy]

When to use M and m? Does it matter?

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.