# How do you prove the statement lim as x approaches -1.5 for # ((9-4x^2)/(3+2x))=6# using the epsilon and delta definition?

You can't.

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

See the explanation.

We are now prepared to write the evidence:

Proof

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

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 -1.5 is less than delta, then the distance between the limit of the function and 6 is less than epsilon.

Let's start by finding the limit of the function as x approaches -1.5. We can simplify the function by factoring the numerator:

(9 - 4x^2) = (3 - 2x)(3 + 2x)

Now, we can rewrite the function as:

((3 - 2x)(3 + 2x))/(3 + 2x)

Canceling out the common factor of (3 + 2x), we get:

3 - 2x

To find the limit as x approaches -1.5, we substitute -1.5 into the simplified function:

3 - 2(-1.5) = 3 + 3 = 6

Now, let's proceed with the epsilon and delta proof. We want to show that for any epsilon greater than 0, there exists a delta greater than 0 such that if the distance between x and -1.5 is less than delta, then the distance between the limit of the function and 6 is less than epsilon.

Let epsilon be greater than 0. We need to find a delta such that if |x - (-1.5)| < delta, then |(3 - 2x) - 6| < epsilon.

Simplifying the inequality, we have:

|-2x + 3| < epsilon

Since we want to find a delta, we can assume that delta is less than 1 (to simplify calculations). Therefore, we can write:

|x + 1.5| < delta < 1

Now, we can manipulate the inequality to find a suitable delta:

|-2x + 3| < epsilon |-2(x + 1.5)| < epsilon 2|x + 1.5| < epsilon |x + 1.5| < epsilon/2

Since we assumed delta < 1, we can say that delta = min(1, epsilon/2).

Therefore, for any epsilon > 0, we can find a delta = min(1, epsilon/2) such that if |x - (-1.5)| < delta, then |(3 - 2x) - 6| < epsilon. This proves the statement using the epsilon and delta definition.

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 #lim (3^(x+1)-2^(x+4))/(3^(x-2)+2^(x-1)+6)# as #x->oo#?
- How do you use the definition of continuous to prove that f is continuous at 2 given #f(x) = x^2 -3x +5#?
- Evaluate the limit by using a change of variable?
- How do you find the limit of #(1+3/x)^(2x)# as x approaches negative infinity?
- How do you find the limit of # (8x^2)/(4x^2-3x-1)# as x approaches infinity?

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