How do you prove that the limit of #x^2 = 0# as x approaches 0 using the epsilon delta proof?
Proof:
By signing up, you agree to our Terms of Service and Privacy Policy
To prove that the limit of x^2 is 0 as x approaches 0 using the epsilon-delta proof, we need to show that for any given positive epsilon, there exists a positive delta such that if the distance between x and 0 (denoted as |x - 0|) is less than delta, then the distance between x^2 and 0 (denoted as |x^2 - 0|) is less than epsilon.
Let's proceed with the proof:
Given epsilon > 0, we need to find a delta > 0 such that if 0 < |x - 0| < delta, then |x^2 - 0| < epsilon.
We start by assuming 0 < |x - 0| < delta. Since x is approaching 0, we can assume that |x| < 1 (we can choose delta to be less than 1).
Now, let's consider |x^2 - 0| = |x^2| = |x| * |x|. Since |x| < 1, we have |x| * |x| < 1 * |x| = |x|.
Since we want |x^2 - 0| < epsilon, we can conclude that |x| < epsilon.
Therefore, if we choose delta = epsilon, then for any 0 < |x - 0| < delta, we have |x^2 - 0| < epsilon.
Hence, we have proven that the limit of x^2 is 0 as x approaches 0 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.
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 # [ ( t^2 + 2) / (t^3 + t^2 -1) ]# as x approaches negative infinity?
- How do you find the limit of #1/(2x+sinx)# as x approaches #oo#?
- What is the value of #lim_(x->0) (1-cos(mx))/(xsinx)#?
- What is the limit of #((64x^2 +x)^(1/2) -8x)# as x approaches infinity?
- How do you find the limit of #x/sinx# 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