Left-handed limit of a rational function

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Homework Help Overview

The discussion revolves around finding the left-handed limit of the rational function (1+x²)/(4-x) as x approaches 4. Participants are tasked with proving this limit using the formal definition of limits.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the behavior of the function as x approaches 4 from the left, noting that it approaches positive infinity. There are attempts to establish inequalities to demonstrate this limit, with some questioning how to choose appropriate values for m and M in their proofs.

Discussion Status

There is active engagement with various approaches to the problem, including discussions on the significance of the values m and M. Some participants have provided insights into the inequalities that need to be satisfied, while others are still seeking clarity on the definitions and their implications.

Contextual Notes

Participants are navigating the constraints of the problem, including the requirement to use the formal definition of limits and the need to avoid the point where x equals 4. There is also a mention of the function's behavior approaching negative infinity at other points, which adds complexity to the discussion.

lep11
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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)?
 
Last edited:
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lep11 said:

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##?
 
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:
lep11 said:
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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