Left-handed limit of a rational function

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  • #1
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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>xm then (1+x2)/(4-x) >m?
But at some point the same function approaches negative infinity so do a choose xm=(something, 4)?
 
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  • #2
PeroK
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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>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##?
 
  • #3
Svein
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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.
 
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  • #4
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When to use M and m? Does it matter?
 
  • #5
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So 4 - 1/M < x < 4 ⇒(1+x2)/(4-x) > M >0 Q.E.D Thanks for super fast replies!
 
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  • #6
HallsofIvy
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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.
 

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