# 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.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- How do you evaluate the limit #w/(1/(-1+w)+1)# as w approaches #0#?
- How do you compute the limit of #cot(4x)/csc(3x)# as #x->0#?
- How do you find the limit of #(x ^ 3)(e ^ (-x ^ 2))# as x approaches infinity?
- What is an example l'hospital's rule?
- How do you evaluate the limit #(2tan^2x)/x^2# as x approaches #0#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7