# How do you prove the statement lim as x approaches 9 for #root4(9-x) = 0# using the epsilon and delta definition?

See explanation.

If

such that

then

That's a lot of jumbled text, so let's make use of it.

has been demonstrated.

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To prove the statement lim as x approaches 9 for root4(9-x) = 0 using the epsilon and delta definition, we need to show that for any given epsilon greater than 0, there exists a corresponding delta greater than 0 such that if 0 < |x - 9| < delta, then |root4(9-x) - 0| < epsilon.

Let's start by considering |root4(9-x) - 0|. Since root4(9-x) is always non-negative, we can simplify this to root4(9-x).

Now, we want to find a delta such that if 0 < |x - 9| < delta, then |root4(9-x) - 0| < epsilon. To do this, we can manipulate the expression |root4(9-x) - 0| < epsilon.

We can rewrite |root4(9-x) - 0| < epsilon as root4(9-x) < epsilon.

To proceed, we raise both sides of the inequality to the fourth power, giving us (root4(9-x))^4 < epsilon^4.

Simplifying further, we have 9-x < epsilon^4.

Now, we can choose delta = epsilon^4. If 0 < |x - 9| < delta, then 9-x < epsilon^4, which implies root4(9-x) < epsilon.

Therefore, we have shown that for any given epsilon > 0, there exists a corresponding delta > 0 (specifically, delta = epsilon^4) such that if 0 < |x - 9| < delta, then |root4(9-x) - 0| < epsilon. This proves the statement lim as x approaches 9 for root4(9-x) = 0 using the epsilon and delta definition.

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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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