Proving a limit is false when L does not equal 1

[Whitishcube]

Homework Statement

Show that if $L \neq 1$, the statement $$\lim \limits_{x \to \infty} (1+\frac{1}{x}) = L$$ is false.

Homework Equations

The Definition of a Limit

The Attempt at a Solution

So I've been trying to prove this by negating the logical statement of the definition of a limit; i.e. by trying to prove that
$\exists \epsilon > 0$ such that $\forall \delta >0 \exists x > \delta$ such that $\left|f(x)-L\right|\geq \epsilon$.
I know that when $L=1$ the limit exists; that is no trouble to prove. The problem is that every time i try to find an x that works, I can never make it work in my proof. Am I going about this the right way?

 

[flyingpig]

Use the limit Law to break up your limit

 

[LCKurtz]

Think about it this way. Suppose L > 1. Now you know how to get the function very close to 1, so you should be able to keep it away from L, no?. So think about what would happen if you choose epsilon half the distance from L to 1. And the case L < 1 is similar or you could combine them.

 

[Whitishcube]

So if I choose $\epsilon = \frac{L-1}{2}$, I want to find an x that will give me $$\left|f(x)-L\right|=\left|1+\frac{1}{x}-L\right|=\epsilon=\frac{L-1}{2}.$$

This is kind of where I'm stuck. I'm not sure what the best way to manipulate the absolute value sign. I know I can do it with either the triangle or reverse triangle inequality, but which would be the right direction to take? would it matter?

 

[SammyS]

Use |L-1|/2, in case L < 1.

 

[Whitishcube]

ok, so starting with $$\left|1+\frac{1}{x}-L\right|\geq\frac{\left| L-1 \right|}{2},$$
I get
$$\left| 1 \right| + \left| \frac{1}{x} - L \right| \geq \frac{\left| L-1 \right|}{2} \text{(triangle inequality)}$$

$$\left| \frac{1}{x} - L \right| \geq \frac{\left| L-1 \right|}{2} -1 = \frac{\left| L-1 \right|-\left|2\right|}{2} \geq \frac{\left| L-3 \right|}{2} \text{(reverse triangle ineq.)}$$

So now I'm stuck here at
$$\left| \frac{1}{x} - L \right| \geq \frac{\left| L-3 \right|}{2} .$$

I'm not too experienced with doing algebra with absolute value signs. Have I been doing it right so far?