# How do you prove that the limit of #(3x+2)=8 # as x approaches 2 using the epsilon delta proof?

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

To prove that the limit of (3x+2) as x approaches 2 is 8 using the epsilon-delta proof, we need to show that for any given epsilon greater than 0, there exists a delta greater than 0 such that if 0 < |x - 2| < delta, then |(3x+2) - 8| < epsilon.

Let's proceed with the proof:

Given epsilon > 0, we need to find a suitable delta > 0.

We start by manipulating the expression |(3x+2) - 8| < epsilon:

|(3x+2) - 8| = |3x - 6| = 3|x - 2|

Now, we want to find a delta such that if 0 < |x - 2| < delta, then 3|x - 2| < epsilon.

Since we have control over the value of delta, we can choose it to be smaller than epsilon/3. This ensures that 3|x - 2| will be less than epsilon.

Therefore, if we choose delta = epsilon/3, we can proceed with the proof.

Now, let's assume that 0 < |x - 2| < delta = epsilon/3.

From this assumption, we can deduce:

|x - 2| < epsilon/3

Multiplying both sides by 3, we get:

3|x - 2| < epsilon

Since 3|x - 2| is equal to |(3x+2) - 8|, we have:

|(3x+2) - 8| < epsilon

This completes the epsilon-delta proof, as we have shown that for any epsilon > 0, there exists a delta > 0 (specifically, delta = epsilon/3) such that if 0 < |x - 2| < delta, then |(3x+2) - 8| < epsilon.

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 find the limit of #(1+2/x)^x# as x approaches #oo#?
- How do you evaluate the limit #lim e^x-x^2# as #x->oo#?
- What is the discontinuity of the function #f(x) = (x^2-3x-28)/(x+4)# ?
- How do you prove that the limit # (x^2 - 4x + 5) = 1# as x approaches 2 using the formal definition of a limit?
- How do you evaluate the limit #(7x^2-4x-3)/(3x^2-4x+1)# as x approaches 1?

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