How do you prove that the limit of #(x^2)sin(1/x)=0 # as x approaches 0 using the epsilon delta proof?

Answer 1
The #epsilon-delta# definition of a limit states that #lim_(x->a)f(x)=L# if for every #epsilon > 0# there exists #delta > 0# such that #0<|x-a| < delta# implies #|f(x)-L| < epsilon#.
Using this for a proof, then, we start by taking an arbitrary #epsilon > 0#, and then showing that such a #delta# exists.
Proof: Let #epsilon > 0# be arbitrary, and let #delta = min{1/2, epsilon}#. Note that because #delta <= 1/2# we have #delta^2 < delta <= epsilon#. Additionally, note that #|sin(1/x)|<=1# for all #x in RR#, #x!=0#.
Now, suppose #0 < |x-0| = |x| < delta#. Then
#|x^2sin(1/x)-0| = |x|^2*|sin(1/x)|#
# <= |x|^2 * 1#
# = |x|^2#
# < delta^2#
# < epsilon#
This shows that for an arbitrary #epsilon > 0#, there exists a #delta > 0# such that #0 < |x - 0| < delta# implies #|x^2sin(1/x) - 0| < epsilon#. Thus, by definition, #lim_(x->0)x^2sin(1/x) = 0# ∎
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Answer 2

To prove that the limit of (x^2)sin(1/x) as x approaches 0 is 0 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 - 0| < delta, then |(x^2)sin(1/x) - 0| < 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)sin(1/x) - 0| < epsilon.

First, note that |sin(1/x)| ≤ 1 for all x ≠ 0.

Now, let's consider |(x^2)sin(1/x) - 0| = |x^2sin(1/x)| = |x^2||sin(1/x)|.

Since |sin(1/x)| ≤ 1, we have |x^2||sin(1/x)| ≤ |x^2|.

To ensure that |x^2| < epsilon, we can choose delta = sqrt(epsilon).

Now, if 0 < |x - 0| < delta = sqrt(epsilon), then |x^2| < epsilon.

Therefore, we have shown that for any given epsilon > 0, there exists a delta > 0 (specifically, delta = sqrt(epsilon)) such that if 0 < |x - 0| < delta, then |(x^2)sin(1/x) - 0| < epsilon.

Hence, the limit of (x^2)sin(1/x) as x approaches 0 is indeed 0, as proven using the epsilon-delta proof.

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Answer from HIX Tutor

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.

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