# Check existence of limit with definition

• MHB
• mathmari
In summary, the conversation discusses using the epsilon-delta definition to check the existence of the limit $\lim_{x\to 0}\frac{x}{x}$ and how to continue the proof. The participants also clarify that since $x\ne 0$, the absolute value of $\frac{x}{x}-1$ is equal to $0$.
mathmari
Gold Member
MHB
Hey!

I want to check the existence of the limit $\lim_{x\to 0}\frac{x}{x}$ using the definition.

For that do we use the epsilon delta definition?

If yes, I have done the following:

Let $\epsilon>0$. We want to show that there is a $\delta>0$ s.t. if $0<|x-0|<\delta$ then $|f(x)-1|<\epsilon$.

We have that $\left |f(x)-1\right |=\left |\frac{x}{x}-1\right |=\left |\frac{x-1}{x}\right |=\frac{|x-1|}{|x|}$.

How can we continue? (Wondering)

Hey mathmari!

Since $0<|x-0|$, it follows that $x\ne 0$.
Therefore $\left|\frac xx -1\right|=0$, isn't it? (Wondering)

Klaas van Aarsen said:
Since $0<|x-0|$, it follows that $x\ne 0$.
Therefore $\left|\frac xx -1\right|=0$, isn't it? (Wondering)

Ohh yes! Thank you! (Blush)

## 1. What is the definition of a limit?

The limit of a function f(x) as x approaches a point c is the value that f(x) approaches as x gets closer and closer to c. This can be written mathematically as:
limx→c f(x) = L, where L is the limit.

## 2. How do you check the existence of a limit using the definition?

To check the existence of a limit using the definition, we need to show that for any given value ε (epsilon), there exists a corresponding value δ (delta) such that for all x within a certain distance from c, the difference between f(x) and the limit L is less than ε. If we can find such a δ for any given ε, then the limit exists.

## 3. What is the importance of checking the existence of a limit using the definition?

Checking the existence of a limit using the definition is important because it provides a rigorous and formal way to prove the existence of a limit. It also allows us to understand the behavior of a function as it approaches a specific point, which is crucial in many areas of mathematics and science.

## 4. Can a limit exist even if the function is not defined at the point c?

Yes, a limit can exist even if the function is not defined at the point c. This is because the definition of a limit only considers the behavior of the function as it approaches the point c, not necessarily the value of the function at c.

## 5. Are there any other methods for checking the existence of a limit?

Yes, there are other methods for checking the existence of a limit, such as using the Squeeze Theorem or L'Hopital's Rule. However, the definition of a limit is the most fundamental and reliable method for determining the existence of a limit.

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