# How do you prove the statement lim as x approaches 3 for #(x/5) = 3/5# using the epsilon and delta definition?

If and only in the event that

We can now put an end to our preliminary thoughts and show our proof to the rest of the world.

Thus, based on the meaning of limit, we deduce that

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To prove the statement using the epsilon and delta definition, we need to show that for any given epsilon greater than 0, there exists a delta greater than 0 such that if the distance between x and 3 is less than delta, then the distance between (x/5) and 3/5 is less than epsilon.

Let's start by setting up the inequality based on the definition:

| (x/5) - 3/5 | < epsilon

Next, we can simplify the inequality:

| (x - 3)/5 | < epsilon

Multiply both sides by 5:

| x - 3 | < 5 * epsilon

Now, we can see that if we choose delta to be 5 times epsilon, the inequality holds. Therefore, for any epsilon greater than 0, we can choose delta to be 5 times epsilon, and the statement lim as x approaches 3 for (x/5) = 3/5 is proven using the epsilon and delta definition.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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