# How do you prove that the limit of #(3x − 5) = 1# as x approaches 2 using the epsilon delta proof?

GIven any

We evaluate the difference:

and we can see that:

which proves the point.

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To prove that the limit of (3x - 5) as x approaches 2 is equal to 1 using the epsilon-delta proof, we need to show that for any given epsilon greater than 0, there exists a corresponding delta greater than 0 such that whenever 0 < |x - 2| < delta, then |(3x - 5) - 1| < epsilon.

Let's proceed with the proof:

Given epsilon > 0, we need to find a delta > 0 such that |(3x - 5) - 1| < epsilon whenever 0 < |x - 2| < delta.

Now, let's manipulate the expression |(3x - 5) - 1| < epsilon:

|(3x - 5) - 1| = |3x - 6| = 3|x - 2|

We want to find a delta such that 3|x - 2| < epsilon whenever 0 < |x - 2| < delta.

Since we have control over the expression |x - 2|, we can manipulate it to fit our needs:

|x - 2| < delta

Dividing both sides by 3:

|x - 2|/3 < delta/3

Now, we can choose delta/3 = epsilon, which implies delta = 3epsilon.

Therefore, for any given epsilon > 0, if we choose delta = 3epsilon, then whenever 0 < |x - 2| < delta, we have:

3|x - 2| < 3(3epsilon) = 9epsilon

Hence, we have shown that for any epsilon > 0, there exists a delta > 0 (specifically, delta = 3epsilon) such that whenever 0 < |x - 2| < delta, we have |(3x - 5) - 1| < epsilon. This proves that the limit of (3x - 5) as x approaches 2 is equal to 1 using the epsilon-delta proof.

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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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