Using the epsilon and delta definition to prove limit

In summary, the limit of $\sqrt(x)$ as x approaches 9 is 3. To prove this, you can use the epsilon-delta definition and substitute $x=t^2$ to simplify the limit to $\lim_{t\to3}t=3$, which can be used to prove the result.
  • #1
cbarker1
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MHB
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Find the limit L. Then use the epsilon-delta definition to prove that the limit is L.

$\sqrt(x)$ as x approaches 9

I figure out the first part of the question. the Answer is three. Yet I have some difficulty to answer the second part of the question.Thank you

Cbarker11
 
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  • #2
I'm pretty sure the square root function would succumb to http://mathhelpboards.com/math-notes-49/method-proving-some-non-linear-limits-4149.html of proving nonlinear limits.
 
  • #3
Put $x=t^2,$ so $\displaystyle\lim_{x\to9}\sqrt x=\lim_{t\to3}t=3.$ Since the original limit is positive, we require that $|t|=t.$
So you can prove your result using the latter limit.
 

FAQ: Using the epsilon and delta definition to prove limit

1. What is the epsilon and delta definition for proving a limit?

The epsilon and delta definition is a mathematical method used to prove the existence of a limit of a function at a given point. It states that for any positive value of epsilon (ε), there exists a positive value of delta (δ) such that if the distance between the input value and the limit point is less than delta, then the distance between the output value and the limit is less than epsilon.

2. How is the epsilon and delta definition used to prove a limit?

To prove a limit using the epsilon and delta definition, we must show that for any given value of epsilon, we can find a corresponding value of delta such that the distance between the input value and the limit is less than delta, which in turn guarantees that the distance between the output value and the limit is less than epsilon.

3. Why is the epsilon and delta definition important in mathematics?

The epsilon and delta definition is important because it provides a rigorous and precise way to prove the existence of a limit. It also allows us to make generalizations about the behavior of functions near a given point, which is essential in many areas of mathematics, such as calculus, analysis, and differential equations.

4. What are the key properties of the epsilon and delta definition?

The key properties of the epsilon and delta definition are that epsilon and delta are both positive values, delta is dependent on epsilon, and the output value must be within a distance of epsilon from the limit whenever the input value is within a distance of delta from the limit point.

5. Can the epsilon and delta definition be used to prove the existence of a limit at every point?

No, the epsilon and delta definition can only be used to prove the existence of a limit at a point if the function is continuous at that point. If the function is not continuous, then other methods must be used to prove the existence of a limit.

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