# Left-handed limit of a rational function

## Homework Statement

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

## 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:

Related Calculus and Beyond Homework Help News on Phys.org
PeroK
Homework Helper
Gold Member

## Homework Statement

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

## 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)
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.

Last edited:
When to use M and m? Does it matter?

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

Last edited:
HallsofIvy