Proof that a limit is less than or equal to zero

In summary, the conversation discusses a proof stating that if a function f satisfies certain conditions including being differentiable at an open interval, having opposite signs at non-zero points, and a specific limit condition, then it can be shown that f'(0) is less than or equal to 0. The conversation also includes a discussion on how to arrive at this conclusion by considering a specific value for the limit.
  • #1

Homework Statement



Proof that, If [itex]f[/itex] is a function such that

(1) [itex]f[/itex] is differentiable at (open) the interval [itex]D[/itex],

(2) [itex]D[/itex] includes [itex]0[/itex] and [itex]f(0)=0[/itex], and

(3) for all [itex]x[/itex] in [itex]D[/itex] other than [itex]0[/itex], [itex]f(x)[/itex] and [itex]x[/itex] have opposite signs

Then

[itex]f'(0)\leq0[/itex]

Homework Equations



None.

The Attempt at a Solution



I managed to prove that for all [itex]x[/itex] in [itex]D[/itex] other than [itex]0[/itex]

[itex]\frac{f(x)-f(0)}{x-0}\leq0[/itex]

I don't know how to get from there to the fact that

[itex]lim _{x\rightarrow0} \frac{f(x)-f(0)}{x-0}\leq0[/itex]Any help would be very appreciated. Thanks.
 
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  • #2
If for some function [itex]g(x)[/itex] we have [itex]lim _{x\rightarrow0}g(x)=L>0[/itex], then can you argue that [itex]g(x)[/itex] must be positive in some (-δ,δ)\{0} by considering a certain ε>0?
 
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  • #3
Okay, I got it. If the limit equals [itex]k[/itex] and [itex]k>0[/itex], then

[itex]\forallε>0\:\existsδ>0 (|x|<δ \rightarrow \left|\frac{f(x)-f(0)}{x-0}-k\right|<ε)[/itex]

implies (for [itex]ε=k/2[/itex]) that

[itex]\existsδ>0 (|x|<δ \rightarrow \frac{k}{2}<\frac{f(x)-f(0)}{x-0}<\frac{3k}{2})[/itex]

But [itex]\frac{k}{2}<\frac{f(x)-f(0)}{x-0}<\frac{3k}{2}[/itex] cannot be true at [itex](0,δ)[/itex], because it would contradict statement (3).

Thanks!
 

1. What is a limit?

A limit is a fundamental concept in calculus that describes the behavior of a function near a specific point. It represents the value that a function approaches as the input approaches a certain value. It is denoted by the symbol "lim" and is often used to solve problems involving rates of change and continuity.

2. How is a limit less than or equal to zero?

A limit can be less than or equal to zero when the values of the function approach zero as the input approaches a certain value. This means that the function is getting closer and closer to zero as the input gets closer to a specific value. This can also be seen graphically, as the graph of the function approaches the x-axis at the point where the limit is being evaluated.

3. What is the significance of a limit being less than or equal to zero?

A limit being less than or equal to zero can indicate that the function is either approaching zero or becoming negative as the input approaches a certain value. This can have important implications in real-world applications, such as in physics and economics, where negative values may represent a decrease or decrease in a quantity.

4. How is a limit less than or equal to zero proven?

A limit being less than or equal to zero can be proven using various mathematical techniques, such as the Squeeze Theorem, the Intermediate Value Theorem, or the definition of a limit. These techniques involve using algebraic manipulation, logical reasoning, and the properties of limits to show that the function approaches zero or becomes negative as the input approaches a certain value.

5. Can a limit be less than or equal to zero at multiple points?

Yes, a limit can be less than or equal to zero at multiple points. This means that the function approaches zero or becomes negative at more than one point as the input approaches a certain value. This can be seen graphically as the function crosses the x-axis at multiple points or approaches zero from different directions.

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